Where exactly are you located in Texas?

I'd suggest that you plug Floratam St Augustine for the sunny area depending on where you live. Very drought hardy. You can just spray round up on bermuda and plug Floratam there. Floratam will fill in fast.

With proper mowing and watering practice, yes, you should have mostly St. Augustine lawn.

Agree. I used to cycle back and forth between St Aug and bermuda in my natural driveway/parking area. By withholding water I could easily transition to bermuda. By adding water and mowing high, it moved right back to St Augustine. I've done the same thing in my back yard when a slight miscalculation on watering and drought resulted in 90% bermuda. It's all back to St Aug now.

It is too late to apply any herbicides due to the heat. Wait until next April and apply something like this...

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It's a little on the dangerous side so read the label very carefully. That stuff will severely weaken bermuda and only slightly weaken St Augustine.

As for right now, you can apply alfalfa pellets (rabbit chow) only to the St Aug and the St Aug runners to improve the growth, health, and greenth (??) of the grass.

Floratam is good. It will thrive in the sun and not do too badly in the shady areas. I have a mix of it in highly filtered shade and in full sun. It is so dense in the sun that I can only mow 1/2 the mower width at a time.