Make sure you don't inadvertently scratch the emulsion side while you handle the film.
The clue to your problem is your reference to the film drying "abnormally fast". This tells me that your humidity is very low. Did you wet down your shower area before you hung the film to dry there?
If you are going to squeegee, it is the base side you should concentrate on.
Also, can you hang your film sideways, so the water only needs to travel about 30 mm to reach the edge?