I turn the cold water off at the water heater. When I open a hot water faucet there is cold water coming out. There shouldn't be any water coming out so it seems like there is a cross connection of some kind allowing cold water into the hot water lines.

Is the cold at full volume and full pressure? Valve stem, ball or cartridge? Has it always been like this?

The water is not full volume or pressure, more like turning a faucet on part way open. The water is coming out of the laundry tub faucet which I believe is valve stem.

I turn the cold water off at the water heater. When I open a hot water faucet there is cold water coming out of the laundry tub faucet
This sounds like the problem's coming from a one handled faucet.
Try this, Turn off the water to the water heater and open up the hot w3ater side of the laundry faucet.
You should have a trickle of cold water coming out.
In turn go around to each single handled faucet and shut off then angle stops. When you come to one that makes the water stoip at the laundry faucet you have located the source of your trouble.
I'll need the brand name of that faucet and its function. Good luck, Tom

Thank you for your answer. I have done what you suggest but it still leaks. I am the second owner of this home and I think someone previously installed a thermostatic mixing valve in the attic above the master bath. It seems the hot water in the shower is not as hot as the other faucets in the bath. I am going into the attic to look around.