Moti Gitik proved that all uncountable alephs can have cofinality ω. [*All uncountable cardinals can be singular*, Israel J. Math. 35 (1980), 61–88] I believe this may be the model you're looking for, but I don't know what happens to non-wellorderable sets in that model. Well, according to the abstract copied below, this model is very close to what you have in mind.

> Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, we prove the consistency with ZF of the statement that every infinite set is a countable union of sets of smaller cardinality. Some other statements related to this one are investigated too.