Reddit mentions of Portable Privacy Pop up Tent Toilet Shower Changing Room for Outdoor Camping Beach Camp with Easy Install

Clothes: 5 gallon buckets and a laundry plunger wand.

Or, if you are going to be driving for a while, this will work well if your shocks aren't the best and the road is bumpy... 5 gallon bucket, add clothes/water/soap/spiky drier ball thingies. Put a lid on the bucket and tighten it, put it in the back. Drive.

When you are done driving, rinse, spot check, wring, and then hang dry.

I have a piece of rope going across the van that I used when doing laundry.

Or you can use a laundry mat but that kind of kills the adventure.

Showering and bathing.

Highly love the solar shower bag by Coleman. You might want to get a pop up shower stand but you can totally bath in your vehicle. A tarp on the floor is helpful.

I took a lot of bucket baths too. Fill a small bucket with soapy water (not tons of soap) and wash up. Then rinse in fresh water. I think that bucket was a gallon bucket.

Wipes are helpful also. When we traveled a lot, some days were crotch and pit wash days since we couldn't shower. It happens. These made that easier for us.

You'll need a few items:

Privacy Shelter 1
OR
Privacy Shelter 2

Solar Shower

I'd recommend being conservative with the water use to avoid mud, aside from that, you should be set!