Ok, so yesterday I got Shinobi III and Contra: Hard Corps in the mail. Shinobi III played fine but when I tested Contra out it didn't load up. At this point I began pacing my room while attempting to fight the urge to throw the cartridge against the wall. I then remembered the whole "Swip the contacts with alcohol" trick and quickly ran to storage room to get out the bottom of rubbing alcohol and qu-tips.
After a few brushes (which came up black) and allowing it to dry off, I tried it again and it worked fine. I'm very pleased I don't have to scream at anyone on the phone to get my money back or package it back up again and mail it back.
So if anyone is wondering, yes it works. Just wanted to share that with you guys. Now I'm wondering how to clean the console itself.