For e.g. the continuum hypothesis, there are models which obey the ZFC axioms in which it is true and models which obey the axioms in which it is false. The Gödel sentences are less interesting; any model in which those sentences were false would be a model in which PA was inconsistent. Which sure, can exist (e.g. the "self-hating theory", PA plus the axiom that PA is inconsistent, is a somewhat legitimate set theory with some interesting properties). So saying the Gödel sentences are true was not entirely accurate; rather, it depends which model we're working with. But a model which declares PA is inconsistent doesn't seem like the sort of model that we'd want to do physics with.

You're right that there are "larger" theories that can prove the consistency of PA, e.g. PA + existence of a large cardinal proves the consistency of PA. But, per Gödel, no consistent axiom system large enough to contain PA can prove its own consistency. This is probably a lot less bad than it sounds though; after all, if we have some unknown axiom system T, and we have a proof in T that T is consistent, does that really tell us anything? Because if T isn't consistent, then it can prove anything, including that T is consistent.

Ok, now here's a question: are there any textbooks introducing model theory and talking about these sorts of topics that don't rely on preexisting knowledge of abstract algebra? Not that I don't want to learn abstract algebra, but I don't even know a good textbook to start that from.