Texas had no right of secession. Any concessions to the Republic of Texas made at annexation would have not applied to the state of Texas due to the Supremacy Clause, and Texas v. White in 1869 held no such right had existed.

Texas never had any special right to secession. The annexation resolution is silent on the matter. It says that Texas may be divided into five states in the future, but all those states would remain in the Union. The original treaty negotiated by John C. Calhoun, which failed in the US Senate, prompting the joint resolution, is also silent on the matter.

So, when Texas was considering secession, it was in the same position as all the other Southern states: secession was legally an open question, with plausible arguments on either side and no definite resolution. Once they seceded, it made perfect sense for them to throw their lot in with the Confederacy, as law and public opinion put them in the exact same boat.

Since then, of course, the legal question has been resolved by force of arms and ratified by the Supreme Court in Texas v. White (1869).