I do have a list of suggestions, some long but the short list is:
Public Works Administration (PWA) not wpa. They funded channel improvements in the Brownsville area and the Lower Rio Grand Floodway project. Don't forget about the railroads, oil, and West Texas.
Do not forget about the LCRA projects to include the Buchanan Dam completed in 1936 - takes several years of surveys, studies, design so 1933/34 time period would fit in.
You would have to dig deep to find any payroll records. Also, the Corps of Engineers had stuff going on in the Valley and Galveston. You have come to the right place to ask questions and hope someone can provide the missing piece.
Would be helpful if you knew what County he was working in or the postmark from your grandmothers letter.
I wish you the best and maybe someone here can guide you.
Thanks so much. These are sources I had not thought about.