While in Austin, we went to the capital building, which is impressive, the dome is slightly bigger than the US capital. What was nice is you can just walk in (though a metal detector) and go most anywhere you want. The legislature is not in session, so we walked on the House and Senate floor, to the outer offices of the Governor, through the rotunda, through the state library and part of the archive. They had tours, but you could just wander on your own, which is what we did and semi-joined tours when the guide was saying something interesting.
Of note, they have portraits of the previous Texas governors in the rotunda and I noticed there have been 2 female Governors of Texas ... the following states have never had any
Arkansas, California, Colorado, Florida, Georgia, Idaho, Illinois, Indiana, Iowa, Maine, Maryland, Minnesota, Mississippi, Missouri, Nevada, New York, North Dakota, Pennsylvania, Rhode Island, South Dakota, Tennessee, Virginia, West Virginia, Wisconsin.