I think many players roll the lip in over the teeth meaning that the teeth are now underneath a bit skin lower than the lip.
I find that with my embouchure, I don't roll the lip back very much. This means that the teeth are underneath a bit of lip which is much more cushion-like.
Looking in a mirror to see what you're embouchure is like is a very good suggestion.
Do you have a teacher? - If there is anything with your embouchure aiding a sore lip, your teacher may be able to point you in a slightly different embouchure that won't hurt as much.
The mouth does get used to it and any pain will go away. The embouchure needed for tenor is looser than for alto, have you made an adjustment?
I first tried playing a baritone after never even playing anything but an alto and could barely get the bottom octave out. It scared me quite a bit that I could be so rubbish. Then someone told me you're supposed to relax your embouchure much more for bari than alto. I gave it a go and soon low As were honking all around Reading, I even got the position as bari player in my county youth swing band and concert band.