We started off with a similar worm tower thing and it ended up being too small for us. It would be full before the bottom layer was finished, and the tap at the bottom clogged often.
We ended up making another bin out of a plastic storage tub by drilling holes in the bottom and setting it on a drainage tray. The bigger bin worked a lot better and there was hardly any drainage. We would let it fill up and empty in spring and fall. It would seem to be getting full, but as the worms ate everything it magically never filled up.
I agree that your bin looks very wet. I used newspaper shredded to a half inch to one inch strips and any other paper from around the house. If an area of our bin was too soggy, we would mix the paper in until it was not dripping anymore. Nowadays, I like to add the paper directedly into our table top compost
bucket so that it's already a good moisture balance when we add it to the worm bin, which is an old bathtub now.
When we first got worms, it took a few months for the population to increase to fill up the bins with worms. I don't remember how long their life cycle is, but it's at least a few weeks. Have you noticed any eggs or small, young worms? You may need to slow things down until the worms have enough babies to match your composting needs. In any given handful from our bin, we could probably find 20 or so worms of all sizes.
The worms will work faster the warmer they are, so if it's possible to move them somewhere a little bit warmer they might speed up. We kept ours in the furnace room when we lived in Vermont.