We recently moved to Friendswood, and the lawn is going to need a lot of work. Yes, zoysia should do fine, but I would research this turfgrass before planting. Zoysia is a warm-season grass that is a slow grower, but once established, it provides a beautiful, dense lawn. It is touted as drought-tolerant, browning with a lack of moisture but responding well when irrigated. This grass will also brown with a hard frost, but it greens early in spring.
Zoysia is said to be tolerant of various soils, but it needs good drainage.
Zoysia does best with frequent, close mowing and moderate nitrogen fertilization. Watch thatch buildup. Late spring and early summer are ideal times to sprig, plug or sod with zoysia. My morning glories won't bloom. Morning glories bloom best in full sun and in a soil that's not overly rich in nitrogen and that is on the dry side.