The CMB happened everywhere in the universe at the same time[0], so what we're seeing is the light which took 13-point-whatever billion years to get here[1] from some part of the universe in that direction at that younger age.

> My understanding is that the early galaxies still produced light after recombination.

I'm not sure what you're imagining about recombination, because I've not heard any suggestion of any galaxies existing before it, so they only produced light after recombination.

[0] for some definition of the concept, even though relativity is formulated with the assumption that there isn't any good concept of simultaneousness.

[1] from our point of view. From light's point of view, time isn't defined.