In increasingly rarer occasions when I process 35mm films, I wash in the way Andrey described, except with tap water. I simply put the tank lid on and invert agitate between filling and dumping water. What I meant was that with good agitation, shaking off remaining water is not all that important. The thiosulfate concentration in the water is already very very low after two cycles, and the concentration gradient drives diffusion weakly while agitation drives much more strongly. That's what I meant.(posted 8278 days ago)The reason I switch to continual flow washing with 120/220 format is because I want to sit down and get my hands off for a while after shaking a big tank. I think fill and dump method is just as good.
I think fixer washing aid solution often mixes bisulfite or metabisulfite in addition to sulfite to prevent possible mineral deposit with hard water as well as preventing excessive swelling of gelatin. Either way, not a big deal in most cases.
It sounds like you have a problem with grit or something in your tap water. If you have a microscope just take a piece of neg to look at what's the dust-like substance you get on the neg. It might be some mineral deposit or grit. Someone brought in some strangely dustly negative for me to look at, and it was tiny pieces of gelatin came off from the film. Under a microscope, you often get much better idea of what it is!