Zach,
Chances are it's a bad seal around the HVAC box....common problem with our type of cars.
To flush out the drain in the heater box, you'd first have to remove the screen, then with a garden hose flush the box and watch as the water and dirt drain out from the bottom of the car. Actually, I think the drain is behind the fender well.
The other method would be to loosen the fender well/ wheel house and stick a coat hanger, welding rod, etc, thru the drain hole.
I'd suggest having someone run the hose in that area while you're in the car with a flashlight. Don't use pressure, just let the hose run without the nozzle.
Beside the HVAC box, it could leak between the pinch weld by the firewall and corner of the right fender. The windshield channel could have a rust hole...another possible leak location.
To remove the windshield moulding you'll need a hook tool to release the clips...or if you don't care about breaking the clips, you can carefully lift the moulding off, but would have to replace the clips afterward.
Blake did an informative thread on removing moulding clips.
https://gbodyforum.com/threads/how-to-remove-window-trim-with-pics.12757/