I think you mean residency. Divorce requires one to be a resident, but not necessarily a citizen, in order to file. However spousal maintenance (i.e. alimony), child support and modifications thereof can be enforced across state lines (and even international lines) even if neither parties are residents of the state which had jurisdiction over the original order.
But getting back to the subject at hand, the Texas law, if it survives legal muster, would likely only affect Texas residents who seek abortions within or across state lines as well as non-residents who seek an abortion in Texas. The legality of out-of-state facilities themselves would be left to the courts and legislatures of the states where those facilities operate, regardless of how close they are to the Texas state line. Advertising limitations are a 1st amendment issue, and could easily be challenged in a federal court. Commercial speech is protected speech. If other states like Georgia or Tennessee follow, similar circumstances and limitations would apply.
I don't agree with the law, in fact I find it repulsive, but I think your claim of "different state, no jurisdiction" being a firm rule of American jurisprudence lacks substance. There have been plenty of legal rulings disproving that philosophy where a so-called compelling government interest is involved.
Take car crashes. If a resident of state A crashes in state B, the at-fault party can be sued in state A if the has court relevant subject-matter jurisdiction (which most civil and criminal courts do) over the case and personal juridiction over the plaintiff (which every state court does over its residents).