No, the shaft should not be turning while the engine is running, that is, unless something is wrong.
Chances are the starter drive may be bad. The drive end may be just touching the flywheel ring gear. Should be an easy fix.
John, Don't get excited, a stethoscope can hear ALL kinds of stuff goin' on, most of which are NORMAL. Play around & listen to your water pump, generator, distributor, etc. Let me know if you can hear a light BULB, BURNING??? Bill W.