Hi coryd
For starters, let me point out that your 6 gallon tank is overcrowded. In addition to that, your cory is a schooling fish and should have some companions. If you have room, it would help to move him into your larger tank and get him some. That will make the little tank easier to manage and clean.
I do not know how large your fish are in the big tank so I cannot tell if that is over stocked too. If it is, solving that situation will help your problem.
Next, I suggest you increase your water changes to 40% every week, or even better - at least until your fish have recovered - 25% twice a week. If they have finrot, or other kind of bacterial infection, clean water is the single most important thing you can do to stop it.
How good is your water circulation? Low oxygen content in the water is another factor that leads to an increase in harmful bacteria, and which stresses and therefore weakens the fish. Increased water changes will help this too. Remember that water tests only test for certain chemicals in the water. They are not an indication that you have enough oxygen or that you have too many harmful bacteria.
Your temperature is good.
Now, about those bacteria....... There are many kinds and different strains of them. Some will be helped by some medicines, others less so. Since they are fish and not humans who can have expensive lab work done, we have to pretty much go by trial and error with their diagnosis and treatment. Fortunately, whatever is in there is a slow acting one, so the chances of getting rid of it are good.
MelaFix is a mild antibacterial and a good one to start out with, but if it hasn't done anything in 6 days, it's not likely to. Let your water changes remove it from the water. Take care not to overfeed so that leftover food accumulates on the bottom, and keep the temperature constant.
Perhaps these things will help. If not, or if the severity of the infection increases, you will want to move up to a stronger medicine, probably an antibiotic. If so, Kanamycin or tetracycline are good, and so is Maracyn if it is used in combination with Maracyn 2. If you use any of these, do be sure that the infection is completely cured before you stop treatment. This might mean running them through the course of medication twice. This will prevent any bacteria from mutating and becoming resistant to the antibiotics, which is one of the worst things that can happen.
Good luck, and please let me know how your fish are doing as time goes on.