It's not too warm yet. I take a church group out on day trips each Lent and we've seen snakes twice. Non- venomous. Not sure if that's good or bad luck. Since we went on weekends during the earlyish spring/late winter with decent weather, camp sites were signed as full up on arrival Sat AMs and probably Friday nights.. I'd think that if you go out other than weekends, less an issue. As suggested, camping in BLM areas may be a possibility as well. In the past I've camped in the Jumbo Rocks, Sheep Pass and Indian Cove campgrounds. Hidden Valley is convenient, too. It's also pretty close to some of the other points of interest. These are easily reached from the north side, going east on 62 from the I-10 near Palm Springs.. You pass Black Rock Canyon Campground going through Yucca Valley but while it may be available, it seems from looking down on it via Google that it's not very inviting compared to other sites I've been too and it's not in the main park road area. Indian Cove is entered off Highway 62 somewhat farther east than Park Bl. It's pretty convenient to some of the local services/shopping if needed compared to other campgrounds in the main section of the park but also not on the main park road. But it's a pretty interesting area. There may still be some water flowing down Rattlesnake Canyon. There are essentially no services in the Park. No water, no stores, etc. Water is available at the entry Ranger Stations. I'd suggest taking in water jugs, bottled water and enough that you don't have to drive back out to get more. How much you might need would depend on both drinking needs, extra just in case water and what you might need for cooking/clean-up.
The Park website has a lot of information and there are other sites with a lot of info, too. Cell phones don't work in the interipr of the park. Not quite sure where you lose service, they do work in the community of Joshua Tree and near the entry station They don't work at Hidden Valley.