It sounds like it has a thermal fuse in it and shuts down when it gets hot, or it just trips the breaker because it starts to pull to much current to turn. If you pull the motor out and feel any resistance when it spins the bearings are probably on their way out. I am sure there are resistance values listed that can be used to test the windings, but I do not know what those values are.
I am not sure what a motor cost, but if its not to much I would probably replace it if I went to the trouble to pull the old one out.