Oof, good question. Stamped steel frame woofers are notorious for this, and you can just tune them with the mounting screws. I would check the woofer out of the cabinet and see if there is a bias, i.e. pressing one side makes more of a difference than another. If so, then the woofer is faulty, and a new one is in order. If the VC seems centered, then I would look at the mounting surface and gasket. Something seems out of tolerance. If it is a cast frame woofer (thick aluminum) then this should not happen in the first place. Again, check the woofer out of the cabinet. The problem could lie in an improperly routed woofer mount, or a lump of crud under the seal. Make sure that the mounting surface is perfectly flat, and there is not a foreign object (like a Volkswagen) caught under the gasket.
Another possibility is a voice coil misalignment in a woofer with an adjustable magnet. This is a very slim possibility, but I am not familiar with the woofer in this system, so I thought I would mention it.
The fact that you did an R&R on the woofer and it made things worse makes me think that it is either a warped frame due to incorrect torque on the mounting screws (or something under the flange) or a piece of stuffing contacting the cone.
gerG