I learned early on that fixer sticks eagerly to skin eagerly and is difficult to rinse off, but perversely is easy to transfer to onto printing paper (and everything else you touch). So now I always use tongs, colour-coded for the different solutions. However, I think it would be obvious to you if your stains occurred where your fingers go.
Then, at there risk of stating the blooming obvious, whatever makes the stains in your prints shouldn't be there. So either you aren't giving the paper long enough in the stop bath, or you aren't replacing the fixer often enough, or you aren't washing for long enough. The action of the stop bath in the emulsion takes a matter of seconds. But FB paper is completely saturated with developer when you transfer it to the stop bath, and it takes longer for the stop bath to get into the fibre base and neutralise it. So although the indicator may show that your stop bath is not exhausted, it may not have penetrated deep enough to prevent carry over of developer into the fixer, where it will react. I give FB paper at least a minute in the stop bath, with continuous agitation.
It's also a good move to use two fixer baths, both because it's more efficient and because it helps prevent stains. Bath 1 is discarded after each session and replaced by Bath 2. Bath 2 is mixed fresh for each session. I forget the exact chemistry, but the principle is that waste products formed in fixer Bath 1 through developer carry-over (and through actual fixing) are removed by the fresher fixer in Bath 2. Ilford recommends 1 minute total fix time for FB paper in their Rapid Fix, but actually I find this isn't enough to prevent staining when I selenium-tone the prints: so I give 1 min in each bath, and again I rock the dish continuously.
You can buy a test kit to check for residual fixer after the prints are washed, which is worthwhile if considering longevity of the print, But I think you are saying that the stains are visible before washing(?), and anyway as you can already see stains it seems superfluous to test for fixer residue. Better to clean up your process to minimise the load of unwanted waste products in each bath!