I have two large oak trees that cover the lawn area, the grass is just not doing well because half the lawn is in full sun,the other in part sun and another in full shade.

I wanted some ideas that do well in central fl in full sun or part sun or part shade or full shade.

Not looking for a complete ground cover per se, thought some small og might do the trick? or not?

I'll take any ideas. I can't have anything that spreads into the neighbor's lawn(both lawns share grass).

thanks!