I removed my old toilet this weekend and found that the flange is both raised and slanted back to front. The front of the flange is approximately 1/2" above the slab (but even with the tile), and the back of the flange is about 3/4"-1" above the slab (thus a little above the tile). The flange is not screwed down to the slab, just sort of hanging in space (which it won't budge so it seems pointless to screw it to anything). The old toilet seemed to do fine. It was shimmed with a clothespin on one side. It does not appear that there were any leaks, and it looks like that toilet was there for many years.

The old toilet just had a traditional wax ring. Should I continue in the same way and just shim and use a wax ring, or is this a disaster waiting to happen? Is there a solution that does not require a new flange?

Best solution is a throated flange,ie: one with a plastic throat that extends down into waste pipe. If toilet will sit flush ( no pun intended) and level on the floor then bolt down snug. Wax will flow to seal connection. If toiler does not sit level then grout the floor to make a level bed and set the toilet. You may have to use two throated flanges and skewer the toilet down to compress the wax into a seal. Try to find the person who set the abs flange and slap him silly.

Your new toilet should absorb the raised side of the closet ring. Use skinny wax ring w/o flange. Set toilet, gently push on it in downward motion and see if it the bowl touched the floor first. If it remains raised - meaning it touched the closet ring first - than you should shim it up and caulk.

Some plumbers use mixture of grout to set bowl into. They put large blob of sanded grout around the base of the toilet and push toilet into it. Wipe off the excess and let dry. This will hold the toilet in place no problem.