I just got it figured out (with some help from sun's java forums.

Turns out that jars in jars are a very bad idea, as you have to write your own custom class loader to navigate the "multi-leveled" jars and find what you want.

As they say, the easiest solution is the best, and it was in this case too. All I have to do is include the extra jar file for the db driver in the applet tag and it works like a charm:

<APPLET CODE="main.TheGameApplet" archive="signedthisisthegame.jar, mysql-connector-java-3.1.12-bin.jar" WIDTH="1000" HEIGHT="600"></APPLET>