I've been using screens for a while now with no issues, prints go on screen face up.

I do not squeegee. I lean the dripping print on the screen at a steep, almost straight up angle for about 10 minutes to let most of the water run off. Then I put the screen in the rack and let it dry for about a day.

I use a dry mount press to flatten the prints after they are already dry, by putting a print between two clean mat boards when the press is cool... Turning it on for a few minutes (just warm, not hot), then turning it off and leaving it for most of a day.

The prints come out "almost" flat... I have seen some flatter prints come to me in the LFF print exchange. So I know there is a better way.