I should add that the old battery will read a pretty low voltage, but it's not consistent. It's generally 10.xx or 9.xx. All the crazy readings I was getting with my meter appear to be the result of that battery, but as you can tell, my electrical skills aren't the best.

Why wouldn't you get an infinite reading on the battery with it in Ohm mode? I've done it many times with different meters over the years, and it's always read OL.

ETA: Thanks for the comment. Some googling shows what your talking about and how I've apparently bought pretty decent meters over the years. I love learning things like this. It'll help to solve a lot of problems down the road.

https://www.eevblog.com/forum/beginn...ery-terminals/

Thanks again!