As many have mentioned, Cottonwood Visitor Center. It was not a long drive in/out and I would only fill up three 5-gallon water jugs every few days. We're really good at conserving water.

In addition to Cottonwood Visitor Center and Cottonwood Campground for potable faucet water, which are both about 7 miles into Joshua Tree, you can go East on the 10 to Chiriaco Summit exit and there is a Chevron Rest Stop, marketplace and a restaurant with potable faucet fill-up and bottled water as well.

There is a campground near the south visitor center inside Joshua Tree. There is a dump station there as well and potable water all for free as long as you can get into the park

Inside Joshua Tree NP, at the Cottonwood visitor center, located between the visitor center and the Cottonwood campground.

I think we went to the campground in the park to get water and empty tanks.