I have a shower stall with a single control knob that controls on\off (pull
out for cold and push in for off) and hot\cold (left for hot, right for
cold). The temperature setting on the gas-fired water heater is on the
"normal" setting. Other faucets in the house seem to produce a lot hotter
water when they run with only hot turned on. For a year or two now I've had
to keep the combination hot\cold on\off handle almost all the way towards
hot to keep the shower hot enough for a decent shower. I finally removed
the round knob and the escutcheon plate to see what it looked like. There's
a very small retangular cutout in the tile and the only thing that's visible
are the two holes that mount the escutcheon plate. The diameter of the
place where the knob mounts is a little bigger than the diameter of a
nickel, and it appears to have what looks like the same type of fitting as
with a garden hose valve when you remove the wheel handle - it looks simply
an on\off function - I didn't seen anything that looks like other mixing
valves that I've seen in pictures - nothing that looks like any adjustment
screws or the like.

Is this a situation where I simple remove and replace what the control knob
attaches to? When I go looking for replacement parts, what's the proper
terminology?

If it's going to involve removing tile and all that, I can learn to live
with it.

Thanks!