I don't know where exactly you are heading to on your way from Kerch.
If you plan to travel down the coast, the first place where you could stay for the night would be Anapa, I haven't seen any hotels on the road in-between (maybe in Vityazevo, but I haven't been there). In-between the frontier and Vityazevo, there is plain nothing.
Anapa itself is a tourist resort and there are many hotels. We had trouble finding one that would register foreigners, though. We finally stayed at at a very luxurious one (US$ 40,-- for a double room) that claimed it would, but only gave us a stamped piece of paper that didn't qualify as registration. We didn't care as the registration rules require you to register if you stay somewhere longer than 24 h (if that is still the case), which we didn't, so we simply kept the hotel bills as proof that we were constantly traveling. We had no trouble at all when leaving Russia, althogh we had one formal registration in Sochi for the last night.
All of that was in May 2005.