I figured out that one of the gas solenoid valve coils was intermittently opening up when it got warm. Once the dryer cooled off, it would work again for a few minutes. Changing the coil was easy, and something most anyone can do. Troubleshooting the issue required a meter and some basic electrical skills. The part was not expensive, and changing out the coil was easy. If you're having the same issue, and don't have a meter, you could just change out both coils and be done with it.