Spreedin, it was more or less the combination of cleaning the window, and the inside of the weatherstrip that I think led to the problem going away. I used ammonia to clean both, followed by good old fashion Windex.

My next possible solution I considered (and still might) is to apply RainX to the INSIDE of the window, as it's nothing more that "window wax." My dad told me that he used to use window wax all the time when he was a kid to help water bead off of windows, and it probably would act as some mild lubricant between the window and the rubber.