Probably more than one problem...
The one where it turns off as you go to hot, could be the spindle for the anti-scald is stuck. You might be able to free it up if you take it apart.
The symptom of decreasing hot water could just be you have a small hot water heater, or it is getting old and not recovering well. If you can check a sink when the hot is going on the shower, does the sink still have hot or is it cooled off as well? My guess is that it is cooled off as well.
Another reason a WH could have poor reserves is if the dip tube (a tube on the cold water inlet that directs the incoming water towards the bottom of the tank rather than mixing with the hot that rose to the top) could be shot. On some models, these tubes disintegrated, and then the WH would only supply a limited amount of hot water since it was getting directly diluted rather than allowing for the natural accumulation of hot water at the top of the tank.
How big is the WH, and how old is it?