I have a tub faucet mixer that I suspect is breaking down. The shower will be hot but when I switch over to the tub the water sometimes gets warm and then cold. If I wait a 10 minutes and turn it on again it will be hot but not always. I thought it was the hot water running out as I have my water heated by the furnace (forced hot water) but the shower will remain hot while the tub faucet plays these games. This is a round mixer with the lever to switch from shower to tub and another lever to raise the temperature.
Sorry to be so imprecise but that's why I'm writing here.

Is this mixer problem and is this a tricky fix and what do new mixers/faucet controls cost to buy and install via a plumber.