Anything high up will generally be snowed in for much of May. One problem is that that tends to keep campers all lower in elevation which means more crowds down low. This is especially true of RMNP which fills with visitors from the Denver/Front Range area. I'd go elsewhere that early in the season.
For early May, might I suggest a couple of towns from which you could venture out.
1. Salida. Low enough in elevation to be quite pleasant most of the time by early May. Drive up the Arkansas River Valley as far as Leadville and take in the scenery. Lots to do in this area. Day trip south to Sand Dune NP. Head down river and poke around Canon City. You can day trip the Highway of Legends from Salida. Salida itself is quite fun to spend some time in, as in Buena Vista and Leadville.
2. Durango. Because you're on the edge of the desert, there are opportunities to explore here even in heavy snow years. Mesa Verde, Hovenweep, Canyon of the Ancients, these can all be explored from here. The Durango-Silverton Narrow Gauge Railroad typically begins full service the first or second week of May. Silverton is worth a day trip, although the streets will probably still be muddy. Gorgeous scenery all the way to Ouray. Or take the road through Tellurde. Given the choice, Durango might be where I'd go in early May.
To be honest, in early May we're still exploring Utah, although so is everyone else. :(