Guns are not only not allowed in most national parks, but not needed. Most caliber guns you'd carry will do nothing more than piss a bear off. I do a lot of backpacking and most trips you won't see a bear anyway. And if you do see one, consider yourself lucky.
Anyway,....a great place to go in the March time frame would be Joshua Tree NP. There are a number of 4wd roads and plenty of accessible camping. I'm assuming you will be car camping? If not, there are wilderness areas where you can camp anywhere. Take PLENTY of water if you go. JTNP is an awesome place. I'm headed to San Rafael Wilderness next weekend for a four dayer, and that'd be a good place to go, but I don't know about off roading. I'll be doing three days in the Sespe Wilderness in March when you're at JT, but may head to JT with the kiddos the following weekend.