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I'm bit confused. Regardless of expansion, these objects are right now > 13.8B ly away. So the light from them still needs that much time to get here. If light from them has arrived sooner, we would see less redshift and perceive them as a closer. No?

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The light we see is from when the galaxy was closer. As the light left the galaxy and travelled towards us, the universe expanded "between" the photons and the galaxy.

And how do we know that now the galaxy is much further away? Just by the red shift of the light emitted over time?

As far as I understand, yes. The red shift of the light can be measured, which tells us for how long the light has traveled.

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