If there's one accessory that's caused more debate and division among woodwind players, it's the humble 'shove-it' - of which the HW Pad-Saver™ is perhaps the best known.
In case you're unfamiliar with the device it's a long, fluffy stick that you shove down the bore of the instrument after you've finished playing.
The theory is that the shove-it will sit in the bore collecting moisture from the pads, which it absorbs and then slowly releases.

It sounds like a wonderful idea - moisture and pads don't make healthy bedfellows, so any device that will remove moisture from a horn's bore has to be a winner.
But, there are drawbacks. The chief drawback is that although the shove-it collects moisture, it remains in the bore because the shove-it remains in the bore too, possibly leaching moisture back onto the pads.
It's nearest competitor is the long-established pull-through, or swab. This is simply a cloth, or sometimes a chamois leather, to which is attached a weighted string. The whole lot is dropped into the bore and pulled through it, collecting and removing moisture as it goes.

Over the years I've heard a lot of hearsay about the differing methods of moisture removal, and this led me to conducting a hands-on test to see which claims stood and which fell.

Two saxes were tested, an alto and a tenor. shove-its were inserted and the instruments were then examined.
It was hard to spot a pad where the shove-it actually made contact. For the open pads there was no way that the shove-it could touch them, and for the closed pads it was rare to find more than a thread or two of the shove-it fibres touching the actual pad. So I jiggled the shove-its about a bit, twisted them too, and managed to increase the contact area on the pads by a very small degree. That the shove-it was fully in contact with the bore was evident - and to a large degree the fibres encroached into the tone holes. Even so, the amount of shove-it touching the pads was very small indeed - so the theory that shove-its will leach water onto the pads is not proven, neither is the claim that they will draw moisture off the actual pads themselves.

The shove-its were withdrawn and then spots of water were dropped onto them in an arbitrary fashion. The water remained in globular form for quite some time. The construction of the shove-it seems to be a non-absorbent 'fleece-like' material, interwoven with cotton threads. The cotton threads exhibited absorption properties, the fleece did not. Over a period of time the water was absorbed down the cotton thread towards the centre of the shove-it.

A good quality swab was then tested in the same fashion. A plain cotton one and a chamois leather one were tested.
As regards contact area, the swab fared better at the top of the bore - from the bottom bow to about the G tone hole, contact with the bore was patchy.
There's no doubt that the swab has a faster absorption rate with the drop test, particularly the chamois one - but it's by no means an instantaneous process. The swab can only collect so much moisture at a time, so several passes are necessary - and time should be allowed inbetween each pass or you'll end up wiping almost as much water back onto the bore as you removed on the first pass! Drawing the swab through the instrument needs to be a very slow process - and if you're a devotee of the swab the you might like to try testing its absorption rate to give you some idea of how fast or slow it actually is!
I noted that a lot of moisture ended up in the tone holes. This was less evident the more passes were made - but I also noted very little contact within the tone holes.

So the problem appears to be this then; During playing the leather or skin of the pads absorbs moisture and the bore of the instrument gets wet. Ideally you need to remove every trace of water from the instrument when you've finished. The absorption rate of a swab is not sufficiently fast to work at the speed with which it is typically drawn through the bore, and a fair amount of water will be pushed ahead of the swab and escape from the nearest exit - which will undoubtedly be a tonehole - there to sit and slowly evaporate or soak into a closed pad.
The shove-it will also push this water into the tone holes, but then as it is left in it will continue to draw water out of the tone holes and into its centre, rather than leaving the water free to dribble onto the pads when the instrument is placed in its case. However, the shove-it has another advantage.
Most of you will have noticed that some of the pads get a little 'crusty'. This debris is formed from the various fats and sugars that are present in your breath and saliva. With a pull through these deposits will be left to dry and form a crust, particularly in the tone holes, but a shove-it - which sits in the bore, and at least in the base of the tone holes - will slowly absorb them. Not all of them mind, but certainly more than a swab will.
Another advantage of the shove-it is that it exhibits a slight scouring action as it is inserted and removed, which helps to prevent a build-up of crust in the bore, and, to some degree, in the tone holes. A swab cannot do this as effectively - it has more of a wiping action.
If we assume that the shove-it does take up all this goo, then surely after a while it will become clogged, and its absorption will lessen. Therefore it's good practice to wash your shove-it from time to time. Use a hand-washing powder. Washing also increases the absorption potential ( ever used a new towel? Useless, aren't they? ).
It's worth mentioning that the shove-it won't clean a very dirty bore. The scouring action is enough to maintain the cleanliness of a fresh bore, but shoving one down a crusty bore will do neither the bore nor the shove-it any favours.

I've heard it said that shove-its are inclined to shed fibres, which can create problems. This seems to be more of a problem with some makes than others - typically the looser, more open designs will suffer more from this problem. To avoid this, get a quality brand - the HW Pad-Savers are excellent, and I've been very happy with the La Voz ones.
I do see horns with fibres sitting in the bottom bow - and quite often collected on the low Eb key pad. Obviously any fibres on the pad may eventually get onto the seat of the pad, thus creating a potential leak. However, this process takes time - and by the time there's sufficient build-up to affect the seal, the pad is often past its best anyway.
Washing the shove-it from new will free up many loose fibres and help to alleviate this problem.

The swab does a better job of cleaning the bottom bow ( the shove-it can't reach it ), but then it negates this advantage by dragging any crud it collects up into the bore, and in particular to the low Eb tone hole - so it looks like with either method the poor old Eb pad gets a bashing.

This still leaves us with the problem of the wet pads, and apart from drying each one individually there isn't an easy way to solve this problem.
So to sum up it appears that there's no really effective solution to the problem of removing moisture from the bore and pads of a saxophone, or any other woodwind instrument. The swab will remove most of the water in the bore, but does nothing for the tone holes, possibly even worsens the problem in the case of a quick pull through. The shove-it will slowly remove the bore water and some from the tone holes, but still does nothing for the pads ( perhaps Bore-Saver is a better name for it! ).
One unexpected advantage the shove-it has is that it has to live in the bore of the horn - so in effect you're forced to use it because there's often nowhere else for it to go in the case. It's all to easy to forego the pull through when you want to make a fast getaway from a gig ( angry punters, last orders at bar etc. ).

As regards the argument that a shove-it amounts to a damp rag down the bore of the sax, well, water will be left there anyway even if you use a swab - and where do you leave the swab afterwards but in the case - so you still have that damp rag problem. Perhaps the most effective solution is to swab first, use a shove-it, and then find time to leave the case open to allow the remaining water to evaporate in its own time.

As for wooden bodied instruments, I wouldn't recommend the use of shove-its. It's debatable whether they'd absorb any of the wood's natural oils, but they'll certainly become coated with them over time. This will pretty much render them useless.
I also have my doubts about leaving a wet shove-it in a wooden bore ( not to mention a wet swab or mop in the case! ).

A quick mention for some of the related shove-it spin-offs, like shove-its for crooks and mouthpieces.
Forget 'em. Not only are there issues of hygiene, you're much better off simply washing these parts out on a regular basis, though there's some merit in using a crook shove-it to keep the crook clean inbetween washes.
One recent newcomer that's worth a mention is the Hodge Swab. This is essentially a silk covered 'sausage', and it's particularly good for keep the upper bow section of baritone saxes clean. I should point out that it's more of a preventative measure...and using one on a crusty old bari will probably ruin the swab. They don't last as long as a shove-it ( about a year or so ) - so consider them a replaceable item.
HW recently introduced a 'bottom bow saver' for altos and tenors that fits down the bell ( the 'S' shaped mop in the photo ). These won't have much effect on the low B and Bb pads, but could prove helpful in keeping the low C# pad clean. They'll also help prevent a build-up of gunk in the bottom bow, though I wouldn't recommend using one on a bottom bow that's already heavily coated in grime.

Personally I have been more than satisfied with the performance of shove-its in my saxes - and let's face it, the last thing I want to be doing is fixing my own horns ( don't get paid for that!! ). In the end it's down to you, and how convenient you find either method. What is certain is that some sort of drying regime is pretty much essential.