Fin rot appears as black along the edge of the fins usually. Extreme cases will form into the fin just fall away rotting off. It would be beneficial to take some photos/video of your Betta and post them here for diagnoses. Some Betta colouring will marble over time, changing colours or transparency.
There is no medicine for Bettas per say, the issue is companies who do produce fish medicine create it for the wide variety of other species. Betta are unique because they have a labyrinth organ which allows them to go to the surface and breath air – other fish obtain their oxygen from the water. Medicines available on the market today do more harm to a Betta’s labyrinth organ than good.
You’re best arsenal against any disease or health issue is Aquarium Salt (AQ Salt), Epsom Salt and a good water conditioner – the water conditioner should always be used regardless, but the salts you should have handy just in case. Combined with a maintained and stable temperature your Betta will recover and even prevent most issues. You can purchase AQ salt from most pet stores or online, Epsom salt can be purchased from chemists or large supermarkets, for water conditioner I recommend API Stress Coat+ which can be purchased from most pet stores or online. Google is your friend for local or interstate aquarium online stores.
Before you being treating your Betta for anything post pictures/video up so we can confirm if it truly is finrot. The last thing we want to do is needlessly treat a Betta for something it doesn’t have. It’s a waste of money, your time and can lower the immune system and effectiveness of the treatment if it were to actually occur in the future.
I do not use filters myself, Betta prefer calm environments with little to no current. I did use one and the cartridge contained a filter and a bio filter. A bio-filter is the name given to the part of the filter in which biological organisms manifest – beneficial bacteria (BB) such as ammonia bacteria and nitrite bacteria (not the official names, but will suffice). Commonly they are just plastic cylinders or balls with slots in it, in which the bio gunk collects and bb forms.
You can replacement cartridges or dedicated bio filters, which I believe are quite expensive, from aquariums, pet stores or online. Other members with more bio filter experience may be able to assist you on this one though.