I would not yet assume the pushbutton unit has failed. On your 124 chassis, unlike the earlier 123 cars, the PBU is protected against short circuits. Where the 123 PBU would melt to signify a problem, the 124 PBU goes into a kind of a hibernation state - you get full heat out the defrost and side vents, at a rather indeterminate fan speed - neither fast nor slow. The unit is unresponsive to anything but the off button, which will eventually shut it all down.
Did it work normally next time you used the car, or is it always in failure mode?
I'm guessing your problem is a shorted auxilliary water pump. The aux pump runs only when the PBU calls for heat. Even if it's shorted, everything works fine until the call for heat. Then you get shutdown mode. I'm suspecting you were cruising along in the rain, with the a/c running. Due to the low heat load, the interior started to get too cool, so the PBU decided to add a little heat to the conditioned air. Then Bam! (to quote Emeril...) the problem happened.
Try this. Find the aux water pump - it's under the hood. Not certain where on your model - I think it's near the monovalve. Look on the passenger side front fender, a bit in front of the battery. It has water pipes running to and from it

Unplug the aux water pump - it has a pigtail with about six inches of wire, then a plug. See if everything works OK, especially the heating mode.
If so, you've found the problem. Given that you live in a chilly climate, you may want to replace the pump before next winter rolls around. Other than that, you probably won't miss it much.
- JimY