^^This is from Braniif's Oct. 1974 Timetable:
Braniff Flight 18 left at 9:00 am nonstop to Dulles.
The 1974 timetable was the earliest I could find on departedflight.com's , but the start date was much earlier. I'm pretty sure it started while LBJ was still in office.
We have nonstop service to all three Washington DC area airports.
Southwest serves National and Baltimore and United/United Express serves Dulles.
Edited to add - I found it shown in the March 1968 timetable from that website you posted.
Depart San Antonio 8:35 am
Arrive Austin 9:01 am
Depart Austin 9:30 am
Arrive Washington Dulles 1:32 pm
Oddly, that same timetable shows the flight then left Dulles at 1:30 pm (two minutes before it arrived) continuing on to New York JFK arring at 2:33 pm.
The service was on a Boeing 727