I live in Hutto, Texas. This shrub started very small and even though I cut it back, it continues to grow and grow. It also self propagates. It has feathery growths on the ends of the stems which fly off when you shake the stems of brush against the bush. The growths appear in the fall mainly. I've also seen this plant all over central Texas.
It looks like Tamarisk to me, also called Salt Cedar.
If this is what you have, I'd recommend getting rid of it as it is invasive in North America. Getting rid of these can be quite difficult though, as they are fairly tenacious.
The genus Tamarix is composed of about 50–60 species... native to drier areas of Eurasia and Africa.
The pink to white flowers appear in dense masses on 5–10 cm long spikes at branch tips from March to September, though some species (e.g. T. aphylla) tend to flower during the winter.
Tamarix can spread both vegetatively, by adventitious roots or submerged stems, and sexually, by seeds.
This plant is not indigenous, but it is now! Texas is a tough place to grow plants for shade, animal habitat, sound reduction, dust reduction. I have a problem thinking THIS plant is reducing your aquifer in Texas. That is the fault of humans only.
This plant our competes other plants because other plants take more resources to survive.
What is it that you are thinking this plant can do to cause any harm? I love this grey color and the fine texture. I love that this plant needs zero human intervention to thrive. Invasive is a curious term. Invasive to what? To whom? Certainly NOT sucking up water other than for its own needs. Any other plant will take more water, it can use water other plants will not tolerate, too salty.
This plant is only invasive because it can withstand temperature changes, salty soils, drought yet you get to enjoy a bit more privacy, birds have somewhere to go in downpours, have somewhere to make nests, this shrub collects dust as well as sound (from neighbors, roads). Yes, cut the suckers off but there is only one root system. If you decide to get rid of this plant, I would love to know what your other choices will be. Have you had a soil test? Few plants accept salt soils. Are you going to change your soil? Are you able to find a more survivable plant for this area and your yard?
Your Cooperative Extension Service is definitely the place to get information! They KNOW this plant and your area and are very knowledgeable about all the sciences in your area of Texas. About the soils, the wildlife and how to conserve resources. Master Gardeners will be your best friend.
I am a bit weird because I appreciate all life, know the bad bad weeds that actually cause harm after they've been introduced by us humans, and to replace that shrub with anything else tells me you might have trouble. That shrub is indicative of your soils and the chemistry of your soils, your water. To go against your environment's statistics will cost money and time and headache.