Water leaks from my cold water and hot water shower handles (separate) whether the water is on or not. How can I fix this?

your not making sense with you question --state it goes cold in middle of shower --but state it will stay hot or cold forever ----if its the latter continued hot till you shut off water that normal working if the unit shuts down during shower then you have a problem ----unit is set at one temp and runs at that temp and you use shower 1 handle / 2 valve what ever you own to adjust temp their are really only 4 reasons stops making hot water ----2 are usually well water supply --incoming water pressure /flow or to cold ----other are gas valve -or sensor so i would look into your shower handles since almost all have adjustments and anti scold safety

No, air can't get trapped in there. I don't think you did anything wrong. It sounds like the valve or internal parts are worn more than new washers take up for. Tightening down the valve just reduced travel for valve to open fully. Can't think of a fix; where's that joe da plumber?

If I understand correctly, you have single handled shower faucet.
Normally there is a wait for hot water to arrive at faucet.
But now hot water is already at shower faucet.

Single handled faucets have a mixing area inside valve where hot and cold come together before going into sink.
This mixing valve can have crossover where the hot and cold can pass through to each other even when faucet handle is off. When single handle faucets get old, this can happen.
How it happens:
For example another faucet is turned on inside house, and pressure differential between hot and cold causes hot water at shower handle to pass through mixing area and into cold water line.
So if another faucet has been run, then that lets hot flow to faulty single handled shower faucet.
To confirm this is the problem: turn on other faucet and put ear against shower faucet to listen for moving water.
This condition may also signal there is a water leak.
The problem is not harmful, but will draw cold water into water heater each time, and is energy wasteful.
Replace faucet or have plumber repair problem.

Replace valve canister- no construction- just remove after taking time to shut off water- a few screws and the job takes 15 minutes. I take for granted you have good water hot/cold to the fixture. This fix will cure most temperature balance problems.
The single lever system is a EZ care system. Note the canister can malfunction faster when exposed to untreated hard water.

what brand of shower handles do you have and do you have just 1 temp handle or is it separate hot and cold?

yes they do,part #1224

sometimes there are orings or gaskets that get tore in assembly one of those may be bad,or if it is around the threds you might try wraping teflon tape around the thred once or twice.hope this helps

The plumbers screwed up when they installed your showers. There are two connections on each shower. one for hot and one for cold. they connected the two hot water lines together with a T fitting and then ran a line to the main hot water line and tied it in. Then they connected the two cold water fittings with a T and ran that line to the main hot water line and tied it in there instead of into a cold water line like they were suppose to. This is the only possible logical solution to this problem. They tied your cold water connections to both showers into a hot water line instead of a cold water line.

make sure when you turn on the hot and cold shut off valves all the way and turn the moen on after that. Some times I would get that problem if I turned one(either hot or cold) off then tried the shower.

you need to change the cartridge