Why is hot water coming out of the faucet but the shower is ice cold? (Diverter not on spout)
If you get warm water out of the spout and cold water out of the shower head, then I sounds like you've isolated the potential problem to the diverter (only other thing that comes to mind is some kind of water pressure based issue, but I can't think of where that would happen on the hot water side only since you have separate hot/cold knobs).

Since this is after the hot/cold knobs, you can probably fix this problem without even shutting off the water, just place something over the drain (don't want to lose a screw), disassemble the diverter, see if it's just an o-ring or dirt, and if not, replace the entire part (take the old one to the hardware store to find an exact match).