2001 Honda Civic ex, belt burning off

They do make clutch and coil replacements, but you usually need to remove the a/c compressor to remove the pulley, this usually ends up being the bearing that's pressed into the pulley. A good junkyard compressor may be the cheapest solution!

Here's how, and if you have enough room you can just swap a new pulley onto it without recharging the system.
COMPRESSOR CLUTCH OVERHAUL
Special Tools Required
A/C clutch holder, Robinair 10204 or Kent-Moore J37872, or Honda Tool and Equipment KMTJ33939,
commercially available
Remove the center nut (A) while holding the armature plate with a commercially available A/C
clutch holder (B).
Remove the armature plate (A) and shim(s) (B), taking care not to lose the shim(s). If the clutch
needs adjustment, increase or decrease the number and thickness of shims as necessary, then
reinstall the armature plate, and recheck its clearance (see Compressor Clutch Check ).
If you are replacing the field coil, remove the snap ring (A) with snap ring pliers, then remove
the rotor pulley (B). Be careful not to damage the rotor pulley or the compressor.
Dont forget the shims!

I am not 100% positive, but I don't think that would lock up the compressor, there is a pressure relief valve to vent high pressures. A locked compressor is more severe than one with high head pressure. But it is true you should have the cooling fan working! One sign of a poor cooling fan is the A/C will probably not get cold if the heat isnt dissapted with the fan.