If you look at the NHTSA reports for our cars and filter to exterior lighting, you'll laugh. Trust me, the odds are that it was not the oil from your fingers. I'll bet you anything that if you pull it again you will see corrosion or melting on the plug/socket that holds the bulb, just like Cruisin said.
Splicing in a new wiring harness is not rocket science; after a few Google searches I'm confident you'd be more than capable of attempting the procedure. If you can cut wires, strip plastic covers with a tool (aka pull really hard), feed two wires into a common connector, squeeze really hard, and wrap electrical tape, then you are qualified.
I paid $25 to fix my own socket in my car. Feel free to pay an additional $75 for some guy who doesn't give a shit about your car to handle it, though.
Last edited by SlipTap; 12-11-2012 at 09:30 PM.