M.A.
We stayed at the Embassy Suites Hotel in north Austin on I-35 and were happy with it. At the time, my daughter was 2 months old and we were able to put her to sleep in the pack n' play in the bedroom and we were out in the living room area and able to watch TV and talk. They definitely have two separate rooms there. We also told them we had a baby and were put in a room at the end of the hall.
We've stayed at Embassy Suites in different cities and are usually very happy with them. A breakfast buffet is included in the room rate.
There is also a newer Embassy Suites near the Arboretum in Austin.
Best of luck,
M.