I started with microfibres and found scrims much better. Microfibres get soaking very quickly, but scrims seem to last longer. Definitely keep them dry, if you use them wet you will leave a mark. The ones you bought come quite large, if its the ones I buy. I prefer to cut them into four pieces and stitch new hems in them to prevent strands coming out, I find them easier that way. Also yeah, new scrims are a bit useless, so again give them a few cycles in the washing machine.
You should only have to give it a run once though, use a dry pre-washed scrim, use one finger to press it into the edge and run it round. Have a look at the edging, then walk away and come back in 10 mins and see if you can see any marks. Ohh, and do the bottom run last, as that's where all the water gathers, so if you start there your scrim will be soaking and you will drag the water round the window. I actually use a different cloth for the bottom edge because of that.