I made a swab a bit like this one http://www.manningmusic.net/p-98-ds-t-cannonball-dragn-swab-for-tenor.aspx by cutting up a microfibre camping towel, and tying the slices along a piece of strong thread. Put a wooden bead on each end to weight it, so I could drop it down the sax. Made a large one for the sax and a smaller one for the neck and mouthpiece. Just need some absorbent material that doesn't shed lint.
I do something even simpler.
I use a pull-through for the body of my tenor made from a small flannel of the type used for mopping babies faces... and other bits. Washed first, though. I have plenty of spares - all left over from our son.
A length of waxed seaming twine left over from sewing sails years ago (plenty more left, too - but any light line would do) a wine cork cut in half and a drawing pin to fasten the thread to it, a swab hitch around a corner of the flannel and the pull-through is made.
The neck and mouthpiece I clean and dry using a Yamaha cloth which was floating around spare - my ex-wife teaches brass. I screw the cloth in the big end of the neck until I feel it rub against the octave tube. I then screw a thin handkerchief in from the mp end until it goes right through to the other end.
If practicing at home or where water is available I sluice off any reeds I have been using and sluice water through mp and neck separately before using the cloth and hankie.
Incidentally when practicing at home if you sluice your reeds off well and dry them with a cloth and then leave them on a shelf
flat side up then your reeds will dry nicely and stay clean looking. It is putting them away in reed pouches - even if washed and dried with a cloth - which makes them go grey and grotty looking. The grey appearance is probably 'spalting' which occurs in felled timber which hasn't been seasoned properly.
