Ed, I had an evaporator (the coils behind the glove box) fail on a car, and the shop that fixed it gave me a big can of "shaving cream" with a hose on it. The idea is to push the hose into or onto the drain tube (which is usually under whatever pan/coiul you have) and then blow out the shaving cream, which is a product designed to clean the coils without harming them. Since it foams and expands it can '"go uphill".
You might be able to get that stuff at any auto parts
If not, I'd get a pump
sprayer, for garden/deck spraying, and either get a bent nozzle for it, or bend up the nozzle, so you can insert the wand and they spray it up and around the coils. Insecticide sprayers usually have tips like that, to spray inside walls.
Your choice of whatever you think will kill the crud, followed by a good water
wash to make sure it doesn't attack the coils.
Hydrogen peroxide might make a good crud killer. Hair supply places carry it in much stronger concentrations (40%) than the 3% stuff from the drugstore.