Only Hot or Cold Water from Shower and Bathroom Sink

A few months back I noticed that the sink faucet in my main bathroom was scalding hot instantly and would not get colder for at least a minute. I could not figure out why scalding hot water would be coming out the minute I turn on the faucet even if I had the lever on cold. Usually it takes faucets awhile to pull in the hot water so I found that odd. I tried closing up the hot water valve under the sink a bit to reduce the amount of hot water coming in, but that did not help. Then a few weeks back the shower in that same bathroom started doing something weird also. The water is either as hot as it can go or too cold there is no in between. I have to turn the lever to the hottest or coldest setting to switch the temperature. If I move it in between it stays at whatever temperature it was last on. I turned down the temperature on my water heater so we are at least not getting burned. What I do not get is nothing has changed and the other faucets in the house all work fine. Cold at first until the hot water gets through the pipes and then you adjust accordingly. I am starting here to see if this is something simple.

Thanks,

This a single family home or multi-unit building?
Do you know brand /make of shower valve and/or lav faucet?
Do you have hot water recirculating system and if so can you post a picture of it and your 2 fixtures that are giving you trouble

Turn the one of the stop valves to lav off 100%. See if the shower returns to normal.

Next try this: Do you have any 2 handle faucets anywhere in the house?
Turn off the cold supply to the water heater. Open the hot only on a 2 handle faucet.

You should get no water out of that faucet once the pressure has drained down. A continuous flow would mean you have a cross over somewhere. Could be the shower valve. go listen to all the single handle fixtures (in closed position) and see if you can hear water passing through them.

No 2 handle faucets? Then close the cold stop to a mixing faucet. when you open that faucet you should get no water flow because hot is off at heater.

I'll tell you the same I tell customers, If your valves don't work, I can't properly diagnose your problem.