https://www.nps.gov/jotr/planyourvisit/camping-outside-of-the-park.htm
Joshua Tree is pretty spread out. A series of day hikes may be a good option. There’s some good stops just driving through the park. Around 29 Palms, 49 Palm Oasis is an easy walk.
https://www.nps.gov/jotr/planyourvisit/hiking.htm
To the south, on the north side of I-10 is the cottonwood springs area. I think it’s Lost Palms Canyon you could hike to and camp.
There’s Whitewater River Preserve. An old trout farm. I don’t know if you can park overnight in their lot. I think you can.  You can’t park anywhere on the road going there. There’s actually a series of preserves. Sand to Sea. You could park at their other preserve, Mission Creek, but that’s dirt road. Well graded. I’ve heard that it actually goes up Mission Creek. It was blocked off by private property. The PCT goes through both those areas. I never filtered there except for taste.
Wildlands Conservancy
https://wildlandsconservancy.org/
Theres another preserve near Pioneertown out of Yucca Valley. Pioneertown itself is an old western movie set. I think you can camp there. There’s a little dinner place out there that sometimes has good music. Robert Plant played there once on a Sunday night with their regular band. Paul McCartney. Of course I missed those nights. I did see Eric Burden.
The preserve near Pioneertown is Pipes Canyon Preserve. It’s a nice walk up the creek. Rich in Indian history.
https://visitpioneertown.com/
You can do a loop through Pioneertown coming out at Old Women Springs Road back towards Yucca Valley. On the hill to the south is an Earthworks exhibit on the side of the hill.
https://www.atlasobscura.com/places/rhythms-of-life
49 Palm Oasis
https://wildlandsconservancy.org/preserves/findapreserve