When we turn our faucet to "off" there is a constant drip of water out of the head. We have a single handle,cold to hot as your turn it. The water dripping is hot, usually.
We have switched the shower head and there is the same problem with the new one.

Which parts do we need to replace?

Ummm, maybe a new washer.

Water is turned on and off at the valve. Single handle valves use a cartridge instead of washers. Cartridge needs to be replace. What is the brand of the valve. Show picture if possible.

It is Moen I believe. Would the cartridge be behind the drywall or easily assesable when the handle is removed?

Cartridge is removed by removing handle.

Moen is warranted for life. Call 1 800 BUY MOEN" for replacement cartridge.

Turn off water. Remove Moen emblem on handle with sharp pointed instrument. Remove screw holding handle on , remove handle, remove chrome sleeve, remove clip holding cartridge in place, remove cartridge. If "O" rings are swollen you may need a cartridge removal tool. Replace cartridge. When replacing cartridge do so with it in the open position to let air trapped in the body of the valve out as you push the cartridge in.