Hi Will,
Some people seem to think that the dough should be quite thin and sticky. I disagree. I think it should be soft and maleable but not sticky (like most good doughs). If it's sticky it has too much water in it, in my opinion. Add more flour until it's no longer sticky
My pizza stone is cracked too. I cracked it when I was making keema naans. Liquid from the keema fell onto the stone and cracked it. I can still use it though and do. Perhaps you can do the same (it's far better than using a baking tray)?
Yes, a thick, heavy frying pan will do instead of a tava. I now use a tava just to "seal" the underside of the naan (15 seconds or so) before I transfer it to the over (it's more manageable, that way, I find).