You should really invest in a test kit and a filter for a tank that size, you must be tired of those big water changes by now!
Fin rot has a very distinct appearance--it's usually caused by the edges of the fins being burned by high ammonia, the wound gets infected, and that's where you end up with fin rot. If your fish bite their tails, which is very common in bettas, the bite wounds can get infected and they can get it that way as well. Infected tissue will be black and crusty looking. If the edge of the fin is clear or white rather than black and crusty, it is a good sign. Clear/white tissue means the fin is growing, either repairing damage from a previous wound or the fish is continuing to grow.
You should really get your water tested--if you don't have a kit and can't afford one, you can take a sample of your water to a local fish store/pet store and they will test it for you for free. You should also consider turning the heater up to around 80 degrees, your fish should perk up with more heat.
Here is the kind of test kit you should purchase: http://www.fosterandsmithaquatics.co...54&pcatid=4454
At some point, you should really look into getting a filter and cycling it. Research the Nitrogen Cycle and make sure you have a test kit before you start.