Shut the power off to the pump. They can absolutely freeze. I’ve seen many well pumps doing as you describe. Open the drain plugs and see if you get water out.
Check the impeller. See if it’s frozen with ice. If it is, thaw it slowly with the drain plug out and get it freed up. Just a small chunk can cause what you have going on.
If it froze and broke the seal, you have a toast motor. If it cracked the wet end housing, you can patch it with something to get you by.
I’ve also seen them break plastic drain plugs. You can rig one with a wine cork if necessary.