Several sedum or stone crop plants stay green all winter. Their foliage is also nice, imo. Otherwise, monkey grass comes to mind.

Native sedges make a nice ground cover. I have found that many of the plants labeled as mosquito preventers don't work unless you sit in the plant. When I lived in Houston, our azaleas did not do well in afternoon sun. Morning and dappled were great for them. A nice shrub (comes in a variety of sizes) is loropetalum. Pretty purple and green leaves, pink blooms and evergreen. Tolerant of sun and shade.

I have also not had much luck with the mosquito repelling plants, unfortunately. Which way is the house facing?

You mentioned afternoon sun, so I'm assuming South or West. I'm in DFW, but here, a few small trees that could take that would be a Crape Myrtle, Yaupon Holly (evergreen), Wonderful Pomegranate, and Vitex. I have not had luck with azaleas in that much sun. Knockout roses would do well, the Drift or the original ones, and here they will bloom from March - December if we have a mild Fall...so almost evergreen, haha. Nandinas are evergreen, and those can take sun. I don't have any in sun, but my neighbors have theirs in West sun and they are gorgeous. Rosemary is another evergreen that is very hardy, and doesn't need much water.

Yes, house faces west.

Really wanted to hear positive words on mosquito plants since our son is always outside but I guess I'll just have to make sure we always have bug spray on hand.

Thank you for the suggestions. I'll look into those plants.