Originally Posted by Andy_K

Actually both sets of cleats I was using were Crank Brothers. They both came with pedals. I admit the first set, at least, do look very much like the Shimano cleats that you referenced, but I'm pretty sure they're different.

My plan is to try the shim and see how that goes. If it doesn't take care of the problem, I'll put the old cleats on the new shoes.