The data source is Texas Department of Health Services and the data is up to data as of September 04, 2020.
The downward movement of total number of hospitalizations in Texas continued past seven days, but seems to have slowed a bit:
The three main contributing TSAs (Trauma Service Areas) are still E – Dallas/Fort Worth, P – San Antonio and Q – Houston:
Houston has almost come down to the same level of hospitalizations as Dallas/Fort Worth and all three are going down in numbers.
V – Lower Rio Grande Valley is still the largest contributor, but still coming down nicely. O – Austin is also reducing hospitalizations in a significant manner.
For completeness, I include the data on total beds and icu beds available and occupied, however, the data is incomplete and I am not sure how accurately it describes reality:
Fig. 4 – Available and Occupied Hospital Beds April 12, 2020 – September 04, 2020 
If the data is accurate Texas is not – and was not – in danger of running out of available beds and icu beds.