Wipe them, put them on a plate or wire rack or whatever, and put that in the bottom of the airing cupboard. Remove when dry. They store indefinitely thereafter. Normally they dry in 2 days.
I've just spent most of today on a solo fungus foray, going to various locations, and came back with masses of field mushrooms exactly like those in the picture, plus 5 slippery jack (Suillus luteus), 2 larch bolete, 2 yellow wax caps, and more puffballs than I can count (2 types). I'll eat the puffballs because they don't dry well in my experience (would love to do it), and one or two of the fieldies, and the rest will go for drying.