You can run a pump when dry but no more than a few seconds. However, you can manually turn the impeller without any worry. The issue is that there is no lubricant for the motor seal without water and if it gets too hot, it could damage the seal. However, it takes a good long time before that can happen.

But the symptoms of the motor sound like a bad start capacitor. You can usually find one online or at a local motor shop. But make sure the motor shaft spins freely first.