What could be causing my underground well house to flood?
If you have metal pipes, and the high utility bill would seem to indicate that your well-house filled due to pumping itself full, digging down to fix it would appear to be a logical approach - though I don't know that I'd be "looking for leaks" as much as "having dug the expensive hole, replacing all the pipes while the hole is open." As mentioned in my comment, I'd use polyethylene water pipe rather than any sort of metal in ground contact. The higher pressure grades are thicker wall, which makes them more durable in general, even if your water pressure is not nearly so high.

This part does not really have anything to do with fixing your leak: Having just replaced it, you won't want to change this part, but I'd also take a long look at not having any "electrical control box" in an underground wellhouse that's prone to flooding. That, however, typically requires changing the pump as well (depending on exactly what it is - if it's a "3-wire pump" and the control box has a capacitor/starter for it, you have to change to a "2-wire pump" - if it's just a pressure switch, you may already have a two-wire pump, and relocating the pressure switch might be all that's needed), and 2 .vs. 3 wire pumps seem to be a philosophical divide among well people. It's also more money on top of an expensive hole, and you may be fine if the expensive hole and pipe replacement means the thing does not fill with water again.