Your thoughts are approximately correct. The Godel sentences are of a different character than the continuum hypothesis (CH) because the Godel sentences are si

Your thoughts are approximately correct. The Godel sentences are of a different character than the continuum hypothesis (CH) because the Godel sentences are simple first-order arithmetic statements, while the CH is a higher-order, a.k.a. analytic statement. A Godel sentence can be assigned a truth meaning via Tarski's definition of truth independent of the axiom system in a way that is much harder to do with the CH. Basically a Godel sentence says something about whether a given piece of software terminates when run on an ideal computer (specifically a piece of software that hunts for a proof of a contradiction within a specified axiom system). I'll argue that whether a specific piece of software would halt or not when run on an ideal machine has a definite truth value independent of any axiom system. Whereas CH doesn't really afford such a software interpretation. I do respect the fact that there exist models of PA + ~Con(PA) but these models are non-standard and we don't use such models to reason about software, specifically because they are unsound in this sense.