Sounds like an old Felton mix.. I'm guessing the washer is worn out and your water temperature is to high.. What happens is the washer starts to swell up as the temperature gets hotter and that's why the pressure drops. To remedy this turn the water off to the house pull the mixer out and replace it (about $100). If its a Felton you will still be able to get spare parts for it (wouldn't recommend that though). I would also recommend installing a tempering valve to go easy on the rest of the tapware.

Another reason for this happing is the cold water pressure being higher then the Hot water pressure and so the Hot water gets pushed back through the mixer body. So you either have a mains pressure mixer (which is wrong) or you need to reduce the cold water pressure going into the mixer (another way is using a check valve).  Remove the faceplate so you can access the mixer, you should find small isolation valves on the left and right side of the mixer body, just isolate the cold a bit more (depending on mixer). In both cases I would replace the mixer

Hope this helps.

does the hot water only die on the shower?

No worries!  I think your on the right track. Keep us posted on how you go... All the best with the new house.