Yes, it's possible and likely that debris from the old water heater could have found its way into the shower valve.
I just replaced my 17 year old water heater because I found debris in a few sink aerators, hot water washing machine hose washer screen and a shower valve that was causing drips from the shower head.
I drained the water heater and there was white sandy plastic debris in the water. I suspect the cold water dip tube was coming apart.
Below a picture of some stuff I found in a bathroom sink aerator.
Also I was getting slightly dis colored water when I ran hot water into the bathtub.
With the new water heater all that is gone and bath water is much clearer.