Well, you were all correct. I did a simple test yesterday that yielded the best result possible. I took a lead from my multi-meter out of the contraption and attached it to a good body ground. Then, I turned the ignition to "accessories" and put the other end of the lead onto the ground terminal of the oil gauge with the old grounds still hooked up. Both gauges immediately began functioning. So it was obviously ground.

Now, the only place in this circuit that wasn't soldered and shrink-wrapped was the bulkhead connector. In fact, I had fashioned that connector in one of the extra slots in the bulkhead connector, so it probably wasn't even as good as stock. So I cut the two grounds before the connector and put them to a good dash ground. The gauges function perfectly now.

Thanks for your help, everyone!