Quick rinse in water after fix and tone. Learned this from Fred Picker I believe decades ago.
The laborious first method also does work , but since you are getting stains, it is telling you the wash is not removing ALL the fix. First you need several rinses after fix, hypo clear, then wash. Soaking in a tray in insufficient. The best method without a print washer is with two trays, fill first with water and interleave prints while second tray is filling. Transfer prints to second one at a time, interleave two cycles while first tray is being refilled. Repeat through 8 trays of fresh water and it will take around 30 minutes. I have prints 50 years old that are fine.
Since you are getting stains, your method will show as stained prints at some point. I urge you to change method.