A low pressure area is spinning across the lower-plains, which will produce enough round of severe weather across the plains today. This is in response to a large upper-level trough stationed across the western two-thirds of the country. One circulation, the one currently producing storms, will lift northeastward and weaken. Another circulation will develop in this trough by the latter parts of the week, moving close to the area by the weekend. This will push a surface cool front close to the area, which will enhance the rain coverage and intensity.
South winds will stay in the forecast, which will bring in ample moisture over the next few days. By Saturday, the aforementioned front will begin to work into eastern Texas and northwestern Louisiana. Most of the rain Saturday could stay over Texas, but the front should be close enough to allow for, at the very least, isolated showers and storms. By Sunday, models agree that the front will work into the state, producing good rain coverage and the potential for heavy rainfall. Parts of the area could see 0.5"-2" of rain with this system. Parts of eastern Texas could receive much higher rainfall totals.
Models start to diverge thereafter. The GFS model is pretty progressive with the front, meaning it moves the front through the area and clears us out on Monday and Tuesday. However, the European model, which is another major global model, is keeping the front hung up over the state, which could lead to high rain chances continuing through Monday and Tuesday. For this reason, the forecast for early next week remains uncertain. With the GFS model having a progressive bias, and the European model making more sense given the time of year, I would lean towards rain chances sticking with us through early next week.