Sorry about the worries....
When you found him on his back and 'dead' was he in the water? How deep is the water and can he (or she) easily climb out of it?
This is a mud turtle and probably a Mississippi mud. It should spend most of the time in the water. But the water MUST be shallow. I agree with the vet that it probably has eaten something. Try bits of cooked shrimp next time you have it or well cooked chicken. Put it in the water in front of it and see what happens.
The flaking you mentioned is common when their shells becomes dry.