I think the CH was used just to provide an example of an undecidable statement, not actually demonstrate something weird about the incompleteness theorem. Since the CH cannot be proven to be true or false within ZF (or ZFC), then you are correct, appending it would just provide an addition axiom.
This is not a weird example for that reason. For a given model, every statement is either true or false. Godel's Completeness Theorem says that a first-order theory is consistent if and only if it is true in some model. Therefore, for every undecidable statement there will be a model where it is true and a model where it is false.
These models can look very strange. For example, if ZF is consistent, then by the Second Incompleteness Theorem so is ZF + "ZF is inconsistent". By the Completeness Theorem, a model for this theory must exist. In this model ZF is inconsistent, so there is a 'proof' of a contradiction from the axioms of ZF. However, since we have assumed the consistency of ZF, such a 'proof' must necessarily involve nonstandard integers.
Well, the CH IS an example of a statement that is undecidable in ZF. Once you include is as an axiom, you essentially bypass the need to use ZF to justify it (or prove it). So, just because you are able to include the CH as an axiom, doesn't mean it is not an example of a undecidable statement within a given set of axioms. Does that make sense?
