Joshua Tree is crazy busy this time of year. It takes forever to get into the park during the peak visiting times. We were out there at the main entrance a couple of weeks ago and the line to get in was 2 miles long. We turned around and went to a lesser known part of the park. It would not be unusual for there to be no open camping spots. While the main roads are paved, you can drive back into the desert on dirt roads. They can be bumpy but are usually in fairly decent shape (as far as desert dirt roads can be). It's possible she got lost on one of them. Some of them are a maze. Cell service in JT is super spotty. When I am out there I often turn my phone off so the battery doesn't drain.
Holly sounds responsible and like she keeps in touch. She was in contact with someone to let them know her change in plans. I am a little concerned she hasn't communicated with anyone since then because she seems responsible and said she would contact that person, but maybe she's just enjoying the peace of the desert and not thinking much of anyone looking for her.
I hope they show her picture to every car entering the park. With that many eyes, if she is there they have a good chance of finding her.