This is a little off-topic here, as none of these schools are much like Tulane, so mods may want to move this.
It depends on what you mean by “best,” as I think that could be different for different students. The most highly regarded academically, particularly for STEM and business, would generally be Texas A&M, but A&M is enormous and has a very particular vibe that isn’t everyone’s cup of tea. Texas Tech offers the big-school, Greek-and-football type of experience along with very solid programs in many areas - the location in Lubbock (5+hr. drive from any major city) can be a little too remote for some. U of Houston has a strong honors program and also offers Big 12 sports right in the middle of a giant city. UT-Dallas was once a commuter school but is trying to build a more residential campus, and they now have a significant number of Natl Merit Scholars on campus due to generous scholarships. U of North Texas is nationally recognized in library sciences and music, particularly jazz music, though it’s still kind of a commuter school in some respects.
Beyond that, it’s mostly regional public schools in both the UT and A&M systems that draw students from nearby areas and focus on marketable degree programs in nursing, K-12 education, accounting, etc. - few out-of-staters except for schools in border areas like UT-El Paso (NM students) and Texas A&M-Texarkana (30% of students from AR).