We were in the Mohave Preserve two days ago and are now in Anza Borrego. We had planned to go through Joshua Tree between the two, but the West Visitor Center was so mobbed we got scared off. It's spring break and Joshua Tree is mobbed by college kids and climbers.
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For us, the Mojave Preserve is as good as Joshua Tree, but without the people. We camped (dispersed) up against the New York Mountains near Caruthers Canyon - no one else going in or out. Saturday night we were at Granite Mountain and the 7 or 8 parties did not fill up all of the better dispersed sites.
Anza Borrego is not mobbed - we even got a site in the state park yesterday at 3 pm. Others have pointed out many great dispersed camping spots, many of which we have greatly enjoyed. And desert flowers are blooming!
Consider a stop in the Carrizo Plain on your way down - there may be a super-bloom. There was some color 5 days ago and if it warms up, it should get a lot better.