Comet Goldfish Problems..?

[Lil' fishy]

So I've had two comet goldfishes for about 2 years now.
Today I was doing a complete tank change because the tank was extremely dirty. (I believe that the filter might be broken)
Since it was only going to take a few minutes, I put my goldfish in separate plastic bowls/cups. I can't really describe it.
Anyways, when I was about done, I noticed that one of them was just sitting on the bottom, not movie. It was breathing and all, just not moving.
I put them both back into the tank. And 3 hours later, I check up on them. That same goldfish was also just sitting on the bottom of the tank. I stared at it for a minute and it started to swim again.
I fed them some food, thinking that maybe they were hungry. However, after they were done, the goldfish sat on the bottom of the tank again.

I then noticed that there were some white stringy things on their fins. It's transparent, but doesn't look like leech. The color is a bit like their fins, and it's really thin.
There are also a few white speck on their heads. It's extremely small and translucent.

This has never happened before. Do you guys have any ideas what might have caused this?

 

[J double R]

when you say the tank was dirty, do you mean that the water was brown? also.. you say you think the filter is broken.. what makes you think that? what kind of filter do you have?

the spots and the stringy stuff smells like severe ammonia or chlorine distress. do you take them from the tank water and put them directly in pure new water?

what size is the tank theyre in, and how large are they?

 

[Lil' fishy]

By dirty, I meant that the water was a bit brown and there was a bunch of things floating around in it. It sort of looked like the floating things were day old food. Which then led me to believe that the filter was broken.
My filter has this pipe thing, where water can enter through. It gets filtered through a sponge and goes back into the tank, where the water is free of anything that may have been in it.

Well, I balanced my water first. And like I said, the water was extremely dirty so I had to put them in the balanced water first.

My tank is about 50 gallons, and they're about 4 inches each.

 

[THE V]

When you say "balanced" do you mean you added in pH stabilizer into the water and then add the fish into it? The pH stabalizer does not remove chorine/choramine.

Tap water commonly contains chlorine or chloramine in low concentrations that helps keep bacteria and algae from growing in the pipes. These chemicals are toxic to fish. When you do a water change you need to add in a chlorine/chloramine remover like "Prime". This binds to the chlorine or chloramine and makes in harmless.

How long has it been since you've done a water change? Generally a weekly 25-50% is recommended to keep the water clean. Doing more with goldfish is not a bad idea. I generally do 80% weekly on my goldfish.