Blow out the drain hole at the bottom of the coil(evaporator) in the freezer next time. Also. Remove the plastic grille from under the door,and back cardboard(at bottom as well) panel. Blow all the dust out you possibly can from the condensor. Most likely the problem is a spider/dust bunny in the drain line though. It may take a good while with warm water and a hair dryer to thaw the drain tube as it runs to the bottom of the unit. Then shove compressed air up its.... drain hole... lol. Should fix you right up.
Sorry , I have no access to compressed air .... The other things you mentioned .... Hopefully they can be accessed from the front ....it would be a devils own job to get the thing out of it's slot as the kitchen part was built around it !!