I think focusing on lower elevations is the way to go in spring. Both high Sierra’s and Colorado mountains will likely have snow.

Tucson area is full of climbs I’d like to do besides Mt Lemmon. Kitts Peak, Mt Graham and Madeira Canyon to name a few. There are a lot of other rides that sound good too. Can start getting hot by the end of May but the elevation shouldn’t be an issue.

Mt Polomar in Southern California is a great mountain with two paved ways up.

Moab is a little higher base so acclimatizing a little is probably smart.

Altitude affects everyone a little differently. Spending a couple of days at higher elevations before hard efforts has always helped me. I’d think you could do all the Tucson area climbs without acclimatizing unless you’re susceptible to altitude issues or have other health conditions. Colorado, and Moab, are different since your base elevations are higher.

You have to go when you have the time but the fall is probably better in my opinion since everything should be open and your conditioning will probably be better. Although the fire season the last few years has made that uncertain too.