So, after a few more attempts at connecting my old project, it seems that the motor, the motor shields, and the power source are all working okay. However, when I went back to use my new Arduino, the motors would not turn. The program was compiling fine and running through, just the motors weren't turning. I realized that the code on the old Arduino was uploaded with a different version of the Arduino software. I attempted to upload newer code onto the old Arduino, and the motors no longer worked. I think that either the drivers for the Arduino are not compatible with the Adafruit library, or the version of the Arduino software. Does this seem like a reasonable conclusion?