Bathroom Sink Draining Water Through Air Vent

We noticed our bathroom sink starting to drain slowly and it’s been years sink the drain was cleaned from hair, gunk, etc.. I started by taking the drain stopper out (lever pull type with a ball-bearing) and used a plastic hair snake to pull out the buildup.

Water began to drain from the sink much better after pulling out a good amount of hair and greasy gunk.. but I then noticed water coming out of the exposed air vent and spilling into our utility closet on the opposite side of the wall (Picture 2)

I disassembled the p-trap (Picture 1) and further cleaned out more clogging buildup. After testing more faucet water, the vent still takes some water and leaks it out the other side but seemingly at a lower quantity.

I fear I may have pushed more clog past the p-trap into the drain that goes into the wall at a level angle

Am I on the right track by trying Drain-O or some other clog remover grease next?

I’m also unfamiliar with what air vent to faucet drain connections look like internally. Is it possible that I damaged some sort of piece that prevents water from going up through the air vent when hair snaking?

Any help and guidance is greatly appreciated. Please feel free to challenge my “theory” of what’s going on.