Saprolegnia in Tank

Alright. After setting up my tank last week and beginning to cycle the filter (I will post pictures later), I added three goldfish to help culture it. Unfortunately, I was gone for several days over the weekend and two of the goldfish died. I believe it was starvation that led to this, though it may have been a water quality issue. (Though I haven't been able to get ahold of reliable methods of testing the water yet, I assumed that the non-chlorinated, local well water running in the tap would be reasonably non-toxic.) I found both fish embedded in the tank's java moss and covered in varying amounts of fungal hyphae likely belonging to genus Saprolegnia.
What do I do about this? Both fish have removed from the tank and the third is under observation, but should the water be changed? How much? As these fish were only intended to help prepare the filter for an axolotl, how much should I worry about recurrent infections? How long will the fungus persist in the filter sponges? Will it even be an issue as long as I keep my animal in good health?
Note: The third fish is, at this point, still alive and well.

Saprolegnia is literally everywhere. Not technically a fungus anymore, by the way. It´s now in the class Oomycota.
I wouldn´t be concerned about the pressence of Saprolegnia (because as i say it´s everywhere anyway and should not be a concern for a healthy animal), but rather about the possibility of having introduced other patogens/parasites with the goldfish. They have been known to carry deseases to which axolotls are susceptible.

Since I wont be having the goldfish and the axolotl cohabitate, would there be any way to ensure that the parasites don't persist in the system after I move the goldfish out? (Thank you for the update on Oomycota, somehow I missed realizing that.)

I wouldn't worry about it persisting in the system as it will always be there. Healthy fish and axolotls have a slime coating on their bodies to protect against fungi and bacteria in the natural water systems and it is only when they are stressed that they become susceptible to ill health and infection.

A classic example is white spot in fish. It persists in water EVERYWHERE and AT ALL TIMES but fish only become ill if their immune systems are already compromised due to stress (e.g. poor water quality, inappropriate temperatures, predation threats, etc). If you keep your fish (and axolotl) happy and healthy and keep the water in good quality then you have nothing to worry about.

As for parasites from the fish my only suggestion is that you keep the fish in the tank for longer than you intended in order to make sure the remaining fish stays healthy. Then remove the fish and add the axolotl at least a couple of weeks after. Unfortunately there is no real way of knowing if a goldfish is carrying anything unless it displays symptoms.