Good news, Thor - I went to the pet store yesterday and I got a lot of new things to keep the tank running in order. I did a 50% water change as well, and my fish seem a lot happier. They're swimming around better than they ever were! I did decide to get a higher power siphon, and I tested it out while I was doing the water change and it worked very well. I also got a water test kit for the tank, and I was really happy to see right away that my pH balance, general hardness, and nitrate/nitrite levels were exactly what was recommended for goldfish. I bought a larger filter as well, to be sure that I have one big enough for the 60 gallon tank.
I think my fish are feeling much happier! I haven't noticed any problems in the tank anymore either, so that's a big relief for me. They seem to all be doing well now. I appreciate all of your help, Thor - you might have saved their lives!