Ballot Access News has the details.
My take, as expanded from a comment there.
They're all unconstitutional, all four bills. Article II, Section 2, Clause 3, as amended by the Twelfth Amendment, specifies that electors shall do certain things. Otherwise, the idea of the founders (pardon me for sounding like an originalist) seems to be that electors should otherwise be "unbound." Yes, that was before the rise of political parties, and the myth that the founders had no reason to anticipate such.
Now, in my opinion, HB 771 and HB 1391 point the way to something that might be constitutional. That is if a state political party requires an elector to sign an agreement to remain faithful on pain of removal from service.
As compared to criminal law, certainly, and even compared to fines, unconstitutionality doesn't carry the same weight in torts, as far as I can tell based on my high-level for a layperson knowledge of criminal law and medium-level knowledge of civil standards. But, requiring a political party to undertake a certain action related to the electoral college might itself be iffy, which the latter two bills still do.
So, the latter two bills point the way toward suggested, or possible, or constitutionally viable, action.
And, unfortunately, we've still not had a so-called "faithless elector" in a state that has some sort of punishments for them pursue legal remedies to the Supreme Court to get a final determination on the constitutionality.
Update, Feb. 12: More voting-related unconstitutionality now being peddled in the Lege, to force registration by party, then block party-registered voters from signing access petitions for independents or, especially, for third parties.. Rethugs will certainly support it in the hopes Libertarians can be held below the 5 percent margin in state races and the Constitution Party would never be able to get the signatures necessary for ballot access. Now that Dems successfully recruited a state Criminal Court of Appeals candidate last year to block Greens' 5 percent push, I'm sure they'd back it too. But, it's all unconstitutional.
Texas is already one of the most independent-unfriendly and third-party unfriendly states in the country. We don't need more.