So today we went out and bought a big jug of Nature's Miracle. Tonight, I started in the bathroom - where the stench is the strongest - and set to work. I poured it all over the carpet and now have the fan pointed into the room. I really hope this works! If it helps the smell in the bathroom, then we'll know that the cats were the reason why our apartment reeks instead of something else. (We have other theories - dead animal in the walls, mold, sewer smells coming in through outlets and light fixtures...)
I also want to get Feliway that I mentioned yesterday. We're getting ready to move in a few weeks, and I think that might help while we try to make the move and for when we're in a new place. I don't want to deal with the horrible smell again!
Because of the problems we've been having, we're not going to try and convince some poor friend to take our cats during the weekend of our move. I'm in the process of finding a good place to board them. In the past, I've boarded them at PetSmart's PetsHotel and had a good experience there. Tomorrow we go to the vet, and I'll ask her if she has any recommendations.