Good hunting. 740 FICO should help a lot.
May I suggest looking at new construction. I lucked out while home shopping during the Great Recession. Other buyers were "stealing" all the properties that I wanted; OR, the sellers were offering seriously screwed up sales.
Then, after walking thru a resale in a sub-division that was still under construction, I saw a great model home. AND, the builder guaranteed financing (to qualified buyers) with just a $1,000 deposit. I'll skip the other cool stuff that I got for buying while the market was cratering.
I just want to say that if you look around you may find a builder (like mine, Adams Homes, sorry no projects in Texas) with a similar promotion, a good reputation, a good location and a standard model that meets your wants.
We bought more than 2,500 square feet in a gated community served by the best local public schools for under $225K. YMMV. Good luck.
You might try searching for a local organization like my local home builders and contractors association. You should also see what kind of financial info is available about builders that you are considering. Entrepreneurs and start-ups can be great but a successful builder with a strong record of delivering what they contract for is nice too.
One more thing. If you buy new, ask if they can use storage trusses for your roof. My builder couldn't (not flexible enough to ask their supplier if they could swap storage trusses for the standard design) You might not be able to add living space in your attic crawl space; but storing crap out of sight for free would be a lot easier without having to dodge all the bracing under my roof.