So I got the paste, came home, managed to wrestle the stock cooler out, cleaned up the old thermal paste, applied the new thermal paste, and went to put the old cooler back in.
I had forgotten exactly how balky these old Intel stock coolers can be. In the process of trying to install it, one of the pins (nine-years-old and apparently now brittle) decided to give up the ghost.
I have reserved a new, better cooler and will be pulling the motherboard tomorrow so that I can install it.
After I get back from Micro Center...