What kind of testing kit are you using to monitor ammonia in your tank? Strips are notoriously inaccurate. Whenever you want to run a filter and cycle a tank and not do 100% water changes it's important to invest in a drops
kit for ammonia, nitrite and nitrate
.. Not just ammonia. You should have been testing for ammonia and nitrite daily and doing a water change every time you saw either get to .25ppm. First you would have seen an ammonia spike. Finally ammonia would stay at zero and you would have seen a nitrite spike. Continue testing daily and finally ammonia and nitrite would have been at zero and you would start seeing nitrates. At this point weekly water changes of 20-50% would keep those under control - maybe two water changes a week at that point if your guys are particularly dirty. It doesn't look like you did this.. if you didn't, this may have caused your problem: http://www.bettafish.com/showthread.php?t=47838
Even if the tank has been set up for several months now, I would definitely be testing all three levels with drops test kit just to be sure nothing has gone wonky. Also really your tank is too small to have divided it so small.. I would consider 4 bettas and shrimp in a 6 gallon a definite overstock. I wouldn't have put more than 2 or 3 bettas in that, as each fish should have at least 2 gallons.
You have a beautiful
tank. However, I see your partitions are clear, and if bettas can see each other that can really stress them out. This set up may not work for your guys. You can choose to use more opaque dividers (obviously there need to be holes for the water to pass through but it doesn't need to be that clear) or put lots of plants up around them so they can't see each other. Long term stress on the adrenals can cause this as well.
At this point your girl/guy with the dropsy needs to be quarantined away from the other fish. Something around 1-2 gallons that you can keep heated would be ideal. You can even buy one of those acrylic bowls, fill it half full and float it in the tank. You need to go find some epsom salt - 100% pure magnesium sulfate according to the ingredients. You dissolve 1 tsp per gallon of water in a little cup, then slowly pour this dissolved mixture into the QT tank over an hour so. Also you need to look for a good internal antibiotic like Kanaplex by Seachem or Maracyn Plus (not to be confused with Maracyn or Maracyn II). Do 100% water changes every other day along with the redosing of meds and epsoms for 2 weeks.
Dropsy is usually fast moving, so time is important.