Trying to wire a 240v line-voltage thermostat

Both lines could "look" hot if you are measuring to ground. But one could be coming from the low-resistance heater, and your high-impedance voltmeter can't even tell it's there.

Simple "line voltage" T stat wiring. One wire is L1 (120V potential) from source, one wire is L2(120v potential)source connected to heating strips, then the switch, then L1 source. When the switch closes the 240V circuit is completed. If you have 240V when checking both wires together, and only two wires on switch, there is something wrong with the new fancy switch. If you have three wires coming from the switch then you need a neutral that you probably don't have.

Makes perfect sense, 12/2 cable is a fine choice for a 240V heater. They don't need neutral, so white is re-tasked to be another hot, and should be wrapped with tape to mark it.

I cannot understand what you mean by two 12/2 wires. A 12/2 Cable contains two Wires (other than ground), colored black and white. You probably do have two cables. One goes back to the service panel and is the supply cable, both its wires will be Hot compared to ground. That is the "line" side.

In the other cable, both wires should have low or no voltage until the thermostat is wired and turned on. That is the "load" side.

Hook up the line side, complete, to the line terminals as marked on the thermostat. Pay no attention to the physical position of the terminals on the old thermostat, they are all different. Once the line side is hooked up, test if it practical. Then hook up the load side. Do them separately like this to keep things simple and understandable.