It was a Speak and Spell! Just remembered today.
Put a paperclip or a pin into the centre of the plug end, then use the multimeter probe on that. It's low voltage and low amps so it's not a problem.
I can't remember off hand what the polarity is for the Genesis 2 and Game Gear, it should be listed on the adapter or console or elsewhere online. With the Game Gear it might also depend on which region it belongs to, US or Japan. You mentioned that you had two adapters with the same part number but the plugs were different, it could be that one is Japanese. I'm pretty sure my Game Gear is Japanese which is why it has a different adapter than every other one I previously came across, but I don't have access to it right now so I can't check it to look at the part numbers. I can't be sure which region yours is from based on your description, probably US but it could be Japanese.
Most likely your adapters are still working fine, it's just that if they weren't then it could have caused low voltage elsewhere in the system. You mentioned not being sure if your adapters were working properly which is why I suggested testing them first, just so there's no further doubt about them. It's sort of like having picture or sound issues with a SNES and not knowing if the problem is with the console or the cable, you need to verify the components separately to better isolate where the problems are.
It could be a defective part, it's rare but can still happen with any component. No idea about the quality of those replacement capacitors either as there's a wide range with different manufacturers. Or there's something wrong in the path that supplies that specific capacitor.