I can understand the pain of dealing with the in-car shower.
Though I don't have an Integra, I have an '88 Prelude with a moonroof (and a '98 Civic coupe with a moonroof, but it's never had an issue). Both of my cars have the same design, a moonroof pan that goes nearly to the rear windshield with a drain tube off either rear corner and a drain tube off either front corner in front of the moonroof.
Well, the Prelude got the in-car shower action shortly after I picked the car up. As I understand it, usually the clogged tubes are the rear ones.
Unfortunately in order to reach the tops of the rear moonroof tubes for either car I have (and likely yours as well), I'd need to drop the headliner, which also requires some other interior panels be removed. If the clamps that hold your rear drain tubes to the moonroof pan are anything like the '98 Civic I parted out, they'll be TOUGH to remove (I just cut them off on the Civic eventually, since I was junking the chassis).
Just sort of a hassle, but certainly not one I want to deal with right now in the cold, rainy northwest US.
Fortunately sometimes the clog in the tube isn't very high up.
If you open the trunk and peel away/remove any trim on the sides of the trunk/hatch area you should be able to see the clearish moonroof drain tube running from the C-pillar and down to the bottom of the trunk area where it drains out a hole hidden by the sides of the rear bumper.
I was able to unclog my passenger drain tube through the trunk. The driver one is clogged higher and will wait for the summer.
Here is a walk through on fixing the clogged tubes. Since it's for the '88-'91 Prelude it might be a little different, but this should be able to at least help out.