JL, I concur with both your recommendations! The latter point about letting it dry out completely before closing the case was a revelation to me about a decade ago. I always leave my sax to dry overnight before shutting the case. Ever since I adopted this routine, no more verdigris, red oxidation spots, nothing. Also my pads and sax adjustment seems to last longer. Storing with humidity is indeed the #1 enemy of saxophones.Yes, get one for your tenor. I'd highly recommend a Hodge silk tenor saxophone swab. It will do the job and will also work on the neck and mpc (unless it's a really small chamber mpc):
https://www.amazon.com/Hodge-Silk-Tenor-Saxophone-Swab/dp/B0002F4YFA
Also, after swabbing it out, it's a good idea to leave the sax out on its stand or in an open case for a couple of hours (or more) to let it dry thoroughly (esp the pads) before closing it up in the case. If on a gig, just do that after you get home.
When I got my Buffet Crampon R13 clarinet back in 1975 there were instructions that came with it that said to leave the case open for a few hours after you finish playing. I've been doing that with all my instruments ever since. Good advice. Even after you swab it out there's still moisture in the pads, and in a wood clarinet some remains in the wood pores.JL, I concur with both your recommendations! The latter point about letting it dry out completely before closing the case was a revelation to me about a decade ago. I always leave my sax to dry overnight before shutting the case. Ever since I adopted this routine, no more verdigris, red oxidation spots, nothing. Also my pads and sax adjustment seems to last longer. Storing with humidity is indeed the #1 enemy of saxophones.