I would start doing daily water changes, perhaps 10-15%, then increase it to 25% daily after the first week. Concentrate on vacuuming the floor of the tank. This will remove a lot of organic matter that may be harboring bacteria and other microorganisms.
If the fins are not looking a bit better (not fully healed, but looking firmer, not quite so filmy) after 2 weeks of improving conditions you should go with antibiotics.
Many medicines work by latching on to almost any organic matter in the tank, including the debris on the floor of the tank. To get the maximum benefit it is best to thoroughly clean the tank before adding the antibiotics. Then the antibiotics will find the organisms causing problems with the fins easier.
The 2 weeks of improving conditions might be all that is needed to start healing the fins. If not, it has at least improved conditions so the antibiotics have a better chance to work.