I'm not sure about the voltage issue. I'm sort of electrically challenged. I can do simple stuff but as far as analyzing why you are seeing 24v after the switch is beyond my expertise.
I'm thinking that you have a different seat and set-up than I had with the one I replaced as I think I remember having two motors.
After doing some searching around, I have found a motor that "looks" similar to the one in the photo you provided. Can you see what the markings read? Do you see anywhere that says that it is a "Motorcraft" motor? Who knows, the seat manufacturer may have used a Ford part in their seat assembly.
I've attached the photo below. In examining it, to me, it looks similar but not exactly like the one in your photo. The one in the illustration below is a Motorcraft MM948 seat motor. It looks as if you can buy one for a bit over $100 new.