First, just so you know, no damage is done by doing nothing. Tarnish is actually good because it protects the finish underneath.
I own two silver axes, a matte/polished engravings and keys, 10M from 1946, and a Conn Transitional Sop that is also matte silver.
I have tried the sulfur absorbing strips and find them to be useless. They may in fact absorb sulfur but I have never seen noticeable effects of using them.
The most effective weekly/daily treatment is a nice gentle rub with a dry treated silver cloth (any brand, find it at household stores). Every few weeks I do a gentle hand polish with Hagerty's all silver polish which works well although I'm sure other brands work well too. I also run wool yarn underneath the rods and around the bases of posts etc... I then do a back and forth "shoe shine" action. This works well to remove dirt and oil buildup in hard to reach spots. Q-tips will also work well as Jim mentioned. Others obviously take their instruments apart periodically and do a more thorough polish with the keys off. You can of course do this too, if you want to take your sax apart. It does require balls of steel or a little know how.
Fred, where did you find that silver cloth online? Was it in bulk form or precut?