I don't have either of these trees, but my chaste trees and lysilomas both send suckers up from the base. I don't think it's something you can stop; it's the nature of the trees.

If you can catch them early when very tender, and rub them off with a gloved hand, it seems to work much better than pruning.

How old are they? They might be suckering to protect their trunks from the sun.

I have a tree that suckered badly when it was first planted, but once it developed a shady canopy, it stopped.

I don't think there is any trump card for winning the battle on suckers, it seems like everything that does well here puts them out.

Olive, palo verde, mesquire, desert willow, ironwood, oleander, african sumac, and so many more.