HDN, have you considered that the 2600 you are using has a dirty cartridge port as well? If so, by all means, you may need to clean that, too. I would use a microfiber cloth for a pair of eye glasses, a thin card, and a dab of 91% isopropyl alcohol on the edge of the cloth. It would clean the port like there is no tomorrow. That could also be the problem. Furthermore, never leave your carts in the port of your game systems while they are plugged into an active socket. The electricity can react to the air and oxidize the leads and the cartridge port. No lie, as it has happened to a Jaguar unit of mine once, which now needs to be repaired because of it (I still have a working Jag with CD, though). If you use a power strip, deactivate it if you leave your game stuff plugged in. That way, no electricity can cause the effect of oxidizing the leads. I can't stress that enough! I have learned from pros.
Once it is deactivated, then (and only then) unplug your retro stuff from a power strip. It's safer for your older electronics, and you have less risk of overloading components on your games systems' motherboards. Again, I cannot stress that enough. Just my sage and friendly advice to extend the life of your retro collection. I have been a professional collector of video games for over three (and almost four) decades. Trust me. I am well versed and my collector friends can back me up. You are creating a great collection and I hope you keep it that way so you can enjoy it. Thanks for letting me share. :O)