I was trying to think this through and realized it would be faster and more accurate to just write up a little script. Pretty sure this is accurate:

https://replit.com/@NateWildermuth/Monty-Hall-Problem

Thinking about this was a lot easier once I could look at the code. It seems like what's happening is that when changing doors, you are actually changing to two doors rather than one, giving you 2/3 odds?