Running the subwoofer outside of the cabinet may not show the problem because there is no "back pressure" applied to it. From your description I would look at the surround (foam ring on the outside) and look for cracks or splits. Also, at the bottom of the cone (from the side) there should be a circular support for the cone (called the spider) that may have come loose. Check that too. If neither of those appear to have a problem, then I would suspect that the internal voice coil has a loose wire. This would require a "reconing" of the speaker or replacement of the speaker.
If you can find another speaker that will fit, try that at low volume to see if the problem goes away. If so, replace the original speaker with a new on of similar specs. If not, then we need to look at the amplifier that may be sending distorted sound to the speakers that are accurately reproducing what they are being sent.
Dan