We add
Axiom CUCSCS: The countable union of countable sets is a countable set.
to ZF.
Is every infinite set now Dedekind-infinite?
My work:
When I look at CUCSCS I see no natural path of building an injection of $\mathbb N$ into a given infinite set (no AOC here).
The answer is either yes, no, or not solved (or unsolvable?!?).
No. In Cohen's first model there is a Dedekind finite set of reals, but countable unions of countable sets are countable in that model.
The proof is difficult, indeed, and requires intimate understanding of the construction of the Cohen model.
The idea, in a nutshell, is that if $A$ is a countable family of countable sets, this fact is forced by a condition, so it is witnessed by a finite set of our generic reals. From that we can prove that there is a uniform enumeration of all the sets in the family with the same support. So all the sets are uniformly counted, and therefore their union is countable.
The proof, in fact, shows that a well-ordered union of well-ordered sets is well-ordered. Which is far stronger than just the countable counterpart.
(See also the Howard–Rubin book Consequences of the Axiom of Choice, Forms 9 and 31 are the relevant ones, Cohen's model is aptly denoted by $\mathcal M1$.)