Update #2
I replaced the cartridge slot with one of these (https://www.amazon.com/Sunjoyco-Replacement-Connector-Cartridge-Screwdriver/dp/B07HP2FFDN?th=1 ) against @Tanooki's advice. Probably should have listened to the guy. It had 1500 positive reviews, so it seemed like it should be okay. Popped it in, didn't work, here's my notes post-installation:
I used 3-4 different carts to test it, all of which worked on my Retron. None worked, at all. At least the original cartridge connector worked sometimes, if rarely.
Anytime I'd put a cart in, some of the pins would get a little pushed out the back about a mm or two. I would push them back in, but eventually one of them wouldn't go back.
Seemed like the carts couldn't ever sit just right.
Bear grip! Hard to get carts in and out. Who would have thought?
Current plan:
Reinstall it one more time, might be out of seat (doubt it, I was pretty careful, but its possible).
Look over the board. Are there any common issues I should look out for? I'll mainly be looking for damaged traces and bulging caps.
Send the connector back for replacement. I could have just gotten unlucky. With that many positive reviews, they have to work sometimes.
Sometimes the NES repeatedly turns off, sometimes the screen is just white/gray.
The new connector definitely was trash, so all it really did was add new possible issues to this whole project.
Again, let me know if you guys want any pics or anything.
EDIT
I looked at the new connector and it looks like some of the pins inside got pushed inside. Not sure if it was my doing or if it was just cheap. Either way, it's going back. Still gonna look over the board, depending how that goes, I may try these guys one more time. Seems like it has the best reviews out of all of them, so I'm hoping I just got unlucky.