The real question is, since you do not want to apply ED anywhere, do you want to apply EA to any schools that Princeton SCEA would exclude (i.e. private schools in the US)? If so, then you must choose between applying Princeton SCEA and applying EA at those other private schools in the US.
Looks like the only schools where Princeton SCEA would disqualify EA applications to are Stanford, MIT, Harvard, and Yale. All except MIT have SCEA/REA (MIT has unrestricted EA). So you can pick at most one out of these five to apply EA to. It does not seem like you have a compelling reason to apply EA to any of the others over Princeton SCEA. In that case, applying to Princeton SCEA gives the possibility of having a decision from Princeton early, which can be advantageous in some ways.
Automatic admission to Texas public universities does not necessarily include automatic admission to the desired major, if the major is a more competitive one. Evaluate chances based on the major if that is the case.