I have gone through this in various iterations. I have 8 Zone VI drying screens in racks. I keep them very clean and use an archival washer. No fixer gets on the screens. I have a dedicated darkroom and dry area.
I used to dry fiber based prints face down on screens with no problem. Then for some reason I started to get intermittent problems. The screens would emboss the emulsion of the prints ruining an entire day of darkroom work. Very frustrating and random. Then I tried drying face up but even small amounts of ambient dust can get embedded in the emulsion.
Finally I submitted to Lloyd's rather obsessive methods and have no problems. Check out his website for useful information. I strung up two vinyl covered retractable clotheslines from which I hang the prints with Clipex clamps from Porter's camera store. No problems since then.
Lloyd's Web Site