It could be normal shedding. They do shed their skin & that will look like thin almost clear stuff hanging off them in the water.
Fungus is kind of furry though. Mine had a fungal problem once in his life shortly after I first got him & I did basically what you've mentioned. I took his home apart & disinfected everything, changed the carbon in the filter too. Then I soaked him for half an hour a day in salty water, rinsed him, then made him sit dry for about an hour. It cleared right up before a week was over.
My thoughts at the time were if it doesn't clear up in a week we're seeing the vet but we never needed to do that. 11 years later it still hasn't happened again.
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PHRatz