Try using rubbing alcohol next as xelement5x suggests in this thread "
how to clean the Sega Genesis Pin Connector" if that doesn't yield better results you would need to than dig deeper. Are you good at using hand tools?
Edit: Always better to try and start with a easier simple fix that more involved. If not so good at using tools than maybe it's time to pass the Genesis along to someone who can and shop for a replacement as you put it just don't toss it the trash you might be able to sell it as it is not working it still worth something. In hindsight I should have suggested the use of rubbing alcohol too but just paper has worked for me. Give that thread a good read.