gar,
I'm not sure what "essentially constant" means. I wonder what he would see if he read the voltages with an analog meter (all trouble-shooters should have one, BTW)?
If the voltage is really steady, and he is getting the amp readings indicated:
I would use a manual thermostat (i.e., connect the wires with appropriate jumpers or use a good thermostat) to eliminate a thermostat issue. It may have been wired wrong. Then, verify the thermostat voltage (could be a bad transformer in the unit), and then verify the correct signals are reaching the unit from the thermostats wire.
If the thermostat and LV wire checks out, it is a unit problem (again, if the voltage are really good). But I am suspect of the voltages because changing other loads in the house seemed to aggravate the issue.