I had the same issue - I'd turn on the AC in the heat, and my lips would crack, my eyes would dry out. I bought a humidifier thinking it would help, but it barely made a difference. Then I figured out what was going on.
The air conditioner does dry out the air, but here's the key thing: it's not because it's actively "pulling out" moisture, but because cold air physically can't hold as much moisture as warm air. When it cools down the room, the humidity percentage drops sharply. And your humidifier just can't add water faster than the AC is "removing" it through condensation. It's not a question of power in the direct sense - it's thermodynamics.
Air conditioners with a humidifying function, by the way, are just a marketing gimmick. A regular AC unit can't cool and humidify at the same time by itself - that goes against how it works. Better to just accept that the air will be drier in summer, and either live with it or keep the AC from running at full blast. Or put the humidifier in the next room - it'll work better there since the water won't be immediately removed by the cold air flow.