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I suspect that any and all members of the Board who are next up for reelection can be removed at that election. You simply nominate and elect different Board members. As far as removing directors for cause, that should be covered in the by-laws governing elections. I presume that there is some procedure outlined there.
I doubt that the specific question of removing directors is covered by TX statute. It's not the kind of thing legislatures get involved in. It is viewed as the nitty-gritty every day management of the HOA, and that is left up to the membership. You HOA organization documents are highly unusual if this type of matter is not covered. It it's not in the by-laws, then delve into the CC&Rs.