Probably not helpful for your situation, but I use a hammock. Pack up the hammock, clean and dry, into a compression, dry sack, then take down the tarp separately and pack it into a non-waterproof bag. I was using a nylon bag that came with the tarp, but I just ordered a mesh bag for this job that will hopefully allow the tarp to dry a bit on road (weather permitting).

When I was using a tent, if the tent got wet, it packed up wet. The solution to make sure at the end of the trip, the tent got set up, dried out, and all the dirt was shaken/brushed out of it.

One thing that might be helpful in my set up, is that the compression sack I had been using is the Summit To Sea eVent dry sack: eVent Compression Dry Sack | Sea to Summit
It lets out air, and, theoretically, moisture even after it's all sealed up. That might have an advantage over a standard dry sack that will prevent air and water from passing in either direction, making sure anything you pull out of the sack is just as moist as it went in. But, to be fair, my compression sack has often been stored in my Ortleib, so its waterproofness has not been really tested, and neither can it really "vent" any water vapor when it's just stuck inside a larger dry bag.