I tried to keep this short and sweet, but it didn't really work. We had a goldfish about a year, along with some sort of algae-eater. Everything was fine until we purchased three more goldies (the black/orange ones). After about a week, we noticed them laying around more, then the problems started.

They started nipping at each others fins (or finrot, or both), the base of their fins started turning red (looked like septicemia, only noticed this on the gold goldfish, since the others were dark-colored), and their slime coat started growing in excess (long, slimy strands hanging off the bodies).

As soon as I saw all these problems, I moved the two I had had for a year into a 10-gal hospital tank I set up on the fly, and applied pimafix and metafix (an anti-fungal and antibiotic) to the water. I completely emptied the 20 gal tank (freaking out thinking the water was too dirty and it was my fault), cleaned everything, refilled it, and allowed it to sit for about a day and a half to evap. The chlorine. I woke up yesterday to find my goldfish and one of the new ones dead.

A few minutes ago, my algae eater wasn't moving, but he was breathing. I panicked and moved the three remaining fish back to the freshly cleaned 20 gal tank, but the algae eater didn't make it.

Before I moved them all to the hospital tank, I tested the water, and all the levels were acceptable. PH was a little low (approx 6.7, normally 7.1), and the nitrite, ammonia, and phosphorus levels were all zero (or very close to zero). I did notice that the water in the hospital tank became cloudy after about a day, and now, in a matter of an hour, the crystal-clear water in the clean 20 gal tank is beginning to cloud. I'm sorry I drug this on for so long. I'm so stressed... I feel like I did everything right, but it wasn't enough. Can I save the last two fish?

Can you check the water now and give me the readings?
Ammonia - I'm guessing this will be high.
Nitrite
Nitrate

Please let me know, it seems like the cycle was damaged in the 20 gallon from over cleaning.
And the 10 gallon hospital tank never had a chance to develop the cycle.

You can read about the cycle here Aquarium cycle (https://www.askmehelpdesk.com/other-pets-animals/aquarium-cycle-nitrogen-cycle-77553.html)

When fish are sick, it's extremely hard for them to tolerate an uncycled aquarium due to all the toxins.

I assure you that the fish didn't get sick by your hand.
So many fish stores sell sick fish that unfortunately infect the healthy ones
After you bring them home.

So sorry you're having to go through this.
Please let me know how things are going.

Do a search on "cloudy aquarium water". Suggestions are to use reverse osmosis water which might be available at a aquarium supply store.

I'd might be tempted to use some distilled water without any gravel in the hospital tank. Use chlorine remover anyway.

I think you acted proactively.

I appreciate the help. Unfortunately, the remaining fish passed today. I guess I didn't get the medicine in the water soon enough. I am going to clean and disinfect the tank and all its accessories, re-cycle it, and start over. I will definitely quarantine new fish in the future. Thanks again.