I moved into a house last September with a water softener. I have never had one before. I have a 55 gallon freshwater fish tank with 4 bala sharks and 5 silver dollars and 2 small pictus catfish. My tank is so cloudy I can hardly see the fish. I have never had a problem with cloudy water so I believe it's because of the water softener. I have tried water changes, new filters and some Crystal Clear drops and nothing has worked! Any advice?

Add salt!

Bala sharks, pictus catfish and silver dollars are brackish fish. They need a fair amount of salt in their water.

As for cloudy water.. because you moved and had to empty the tank, the good bacteria in the tank died off. The good bacteria is very fragile and can die off over any reason. To help your tanks cloudiness out, try adding live bacteria to your tank to help cycle it. You can buy pure bacteria by SEACHEM or you can get a dechlorinator that helps boost the good bacteria in your tank along with ammonia, nitrites and nitrates. Its called PRIME and is by the same company. Try not to do any more water changes while you're building up the good bacteria in your tank. Keep an eye on your catfish, if anything goes wrong in your tank, those guys are the first to get sick. I'd almost advise to purchase some Pimafix for any type of fungal infection that is associated with poor water conditions.

How long is the tank set up? It takes about 2 weeks for good bacteria to achieve good levels to consume waste. All the fish mentioned can be rather large, translating in to a large amount of waste for the bacteria to catch up with, so adding the bacteria is a good idea.Also, cut back on feeding!
Second:
Water softeners use salt additives to soften hard water. I am not an expert on "Water Softeners", however because they use salt, I would check the salinity coming out of your tap before adding more to the tank.
I use water softener salt from the supermarkets for my ponds. It's exactly like rock salt minus a clumping agent used in rock salt. Which is detrimental to fish, and a lot more reasonable than aquarium and pond salts.Salt is salt, don't waste your money softener salt is $5 for 40 lbs, aquarium/pond salt is about $2 a lb.
I think lucky98 had a typo, advising that bacteria will raise ammonia,nitrite and nitrate.

Relax and stop changing the water too much, give the bacteria a chance to grow as lucky mentioned.

Are the fish ill? If not, I would wait and see before adding anything.
And test your water parameters. If all is correct wait and the water will clear up in time.

All of these fish are 8 years old. I have never changed the water more than three times a year and never more than 25% at a time. None of the fish are acting sick. Tested the ph in the water and it reads at 6. Not sure what else to do...

My angel fish get the same treatment and my Ph is always low and the water is soft as in their natural Amazon habitat. However, the Bala sharks come from Thailand where its 7.0. You need to raise it slowly but not too slow.
This is where your water softener come in to play.
Your bacteria is not getting a chance to grow due to the soft water and a low ph. If the GH/KH (general hardness/carbonate hardness) is too low your good bacteria cannot survive let alone multiply. To correct this, simply add crushed coral in a fine mesh bag to the filter or place it in the tank. This will raise the hardness of the water and in turn prevent the ph from crashing. You need get an API test kit to check Ph/Gh/KH/ Ammonia/Nitrites and nitrates.
They sell 20 lb bags of crushed coral for salt water aquariums, about $15. Only use about a pint of the stuff.

You have to check the PH daily as the crushed coral will help raise and stabilize the ph. If it gets too high or stays too low, pull out the bag of coral and place more or less in the bag according to your needs.
I would place it in the tank to make this easier on you.
Bacteria do better in higher ph but most fish won't. So you need to be on top of this.

I lost 10 years of koi 18" to 24", due to a ph crash. Long story short the ph was about 5, I kept feeding, the ammonia went to 8, it killed all the bacteria, and in turn the water was like acid on the fishes skin. Lost 10 koi in less than 24 hours. That is why I believe you could have a similar issue. I would cut back feeding to every 3 days to slow up the fish waste. They will not starve.

I had this happen I tried everything until I finally spoke to a fish guy who knew what he was doing he said to leave it alone it took 4 months for the bacteria to even out and it did it was frusterating to wait that long but worth it in the end never had a problem since it just took time so be patient it will clear I too had a water softener and similar fish