Your sega will probably just keep doin this forever. I don't think systems of this generation will ever not work (as long are you aren't a neanderthal... But even then, maybe it will)
Thirty years ago my grandmother was carrying my 1987 nes down some stairs and her sock caught on a nail. She fell down the stairs, broke her hip, two fingers, a toe, shattered her left arm in 9 places around the elbow, suffered a concussion, and was in icu for two weeks, plus a trip later 500 miles north to the Mayo Clinic to get an extensive nuts bolts and plates surgery to save her arm from being amputated.
The NES survived unbroken and still works to this day. Tecmo Super Bowl is still the best football game.
Good ole rubbing alcohol and a q-tip! That's all you need.
Edit: I don't know if you have "q-tips" - it's a cotton swab or cotton stick, would be the generic name. The things you aren't supposed to stick in your ear but that's what everyone does anyway.
I was going to ask if a qtip is just a cotton bud! Also is rubbing alcohol the same as an alcohol-based hand sanitiser, or do I need some specific alcohol-based cleaner?
Don't use hand sanitizer, at least not the gel type, it will leave a residue. You just need like 91% isopropyl alcohol - stuff you would use to clean a wound.
Edit - I looked it up, it's called surgical spirit in the UK.
Hell yeah. Yeah - 25 years of crud builds up. Those contacts just need a lil cleaning and you are good to go usually. Also if you feel so inclined - swapping out the pin slot in the system itself is very easy and cheap. There's no soldering on the NES, not sure about the Sega.... But honestly if just cleaning the pins on the game did the trick, I wouldn't fuss with it.