New Mexico is at least four different states. Divide it into quarters, using I-40 and I-25.
In March, I would be going to the SW quarter, out toward Deming, or touring US-60 into southeastern Arizona. It might start looking like Spring, in March, that far south. Spring in the desert can be beautiful.
Unless I wanted to join the Spring breakers for skiing, snowmobiling or snowboarding, then I'd be going to the NE quarter, which is still a winter sports hotspot in March.
Can't recommend campgrounds in the SW part, that is still on my to do list. My last trip was to the SE quarter of the state, staying in Ft Sumner and Roswell, too long ago to verify the parks are still open or under the same management.