Ok, well now I feel like a fool.
I resoldered all the connections (re-melted them, really), especially the ground ones. The output signals still weren't perfect on the scope, so I didn't think that fixed anything. Came home and tried it on the TV, and BOOM! Perfect. Like, really perfect image on every TV I could find in my house.
I'm not sure which connection / combination of connections was to blame, but it's gone now.
Thanks so much for all your support, Uze!! You have no idea how appreciated it is! Seriously, getting this kind of help and understanding on things these days is super rare!
Now, on to make the controllers and attach the SD card interface!