Yes, the La Plata is small (not raftable, not much fishing). Some (those on the North end of Lime Creek road). Yes, Lower Piedra CG is just off 160 on the west side of the river. Fly fishing will not be good until the rivers go down and clear up. You didn't say how much rafting experience you have so I can't recomend a run to start on. Always wear a life jacket.
None of the surface water will be potable without treatment, bring some water with you for a few days. Not sure which of the Forest Service campgrounds will have drinking water, check their website or stop in or call a ranger station for info. They have good maps too, and road condition reports. Have fun.