We ended calling Frigidaire and they sent a technician a couple of weeks later to install a retrofit part in the housing of the water filter. It worked for a few weeks (the cartridge stopped "kicking out"). However, coincidental or not, the dispenser stopped dispensing water about a month afterwards. Rather than spend money to fix the problem, we ended up getting a new refrigerator because we were frustrated by the whole ordeal.
It shouldn't. Try flushing a few gallons of water through the filter. I think the old clear style filter had instructions to flush the filter after installing. Don't worry about running too much water through the filter. As far as I can tell, the light for replacing the filter is based on time in service, not by gallons/liters of water consumed.