Yes, we eventually unplugged the fridge, but the smoke/burning appears to have been a singularity, not something continuing to draw electricity and heat up if that is a clue.

I finally was able to dig underneath the fridge last night. The white/biscuit connector to the compressor (starter?) seems to be the problem. When I removed it and started disconnecting the wires, one (blue) came out along with the post from the starter which was black. The plastic of the starter around the post was melted and brown and the starter itself rattles like a child's toy. It still had the faint smell of smoke to it so I know that it, at the least, is dead and have ordered one from APP.

I am hoping that it just died and it's death was not caused by something else. I have checked the compressor for continuity and those checks are all good. I do not have the tools to check the capacitor, but I have heard that they do not often go bad.

Open to suggestions on anything else to look for while I am crawling around, but expect the part tomorrow or Thursday and will report on success.