Ok. That is actually not warm enough for bettas, and especially with swings of more than 1-2 degrees. They often get velvet in these conditions. Velvet is definitely curable, but an adjustable heater will be necesary. Betta's thrive at 80. 78 would be acceptable as well, but they shouldn't be kept any lower than that. Consistency in the temp is even more important.
While I can't give you a definite diagnosis, it does sound like velvet. Sometimes as bettas mature they develop a silverish beard, so it's possible that is what you are seeing. Velvet would cause the betta to clamp fins and be lethargic, but so will temps of 75 and lower, so it's really hard to say. You can try to get a pic of it but I find it somewhat hard to diagnose by photo. As velvet progresses, it becomes pretty obvious..you can see the dusting effect usually starting at the gills and will begin to cover the fins and if it gets real bad, down the sides of the body. To treat velvet add an adjustable heater and slowly raise the temp to 80, increasing by 1 degree every 12 hours. I'm not sure how much salt you've already added but ideally you would gradually add 3 teaspoons, 1 each day for three days. Always disolve salt in dechlorniated water before adding to the tank so it doesn't burn the fish. Ideally the fish should be symptom free within a few days to a week and you can start water changing out the salt by the 10th day. You want to do a few partial water changes over a few days to get all the salt out. You don't want to have salt in the water for more than a couple weeks or you can cause other problems. I recommend a Hydor or Aqueon heater, but any adjustable should be fine. Just do not select one with a pre-set temp as they won't raise the temp high enough.
Also if there is ammonia in the tank it is probably not completely cycled. The bio-load of 1 betta in a 10 gal is small so there shouldn't be any ammo in there. I would start by doing a large water change today, and keep a close eye on the parameters. The ammo could be responsible for some fin deterioration you mentioned.