Thanks. As you can tell, it is a home unit, not a car. I was concerned that hooking it up direct might damage the woofer since there's no xover. I imagine that's the case with a tweeter, right?
The Woofer is moving but just distorted.
If the woofer distorts when hooked up directly, there is a problem with the woofer.
A woofer is not harmed by the direct connection, though a tweeter can be if one is not very careful. Typically, tweeters cannot handle much power, and cannot handle much input in low frequencies. Crossovers protect tweeters by filtering out bass frequencies to them.
So, it would appear that you need to replace the subwoofer. However, that does not prove that there is nothing wrong with the subwoofer amplifier, too. You could hook up another woofer to it, which normally would be safe, but sometimes amplifiers have problems that damage speakers. Since the subwoofer was still working (although improperly), it is likely that you could hook up another woofer to the subwoofer amplifier without much danger.
If you have a cheap and unimportant woofer lying about, I would hook it up and try it out to make sure the subwoofer amplifier is still functioning properly. You will want to start with the level turned down quite low, as speakers generally cannot handle as much power when they are not mounted in a cabinet than when they are (due to the air pressure in the cabinet preventing the cone from moving as easily as it can outside of a cabinet).
If it works without distorting at low levels that way, you probably just need to buy another speaker for your subwoofer cabinet.
In the case of rare or valuable speakers, they often can be rebuilt if one cannot find a new one. I do not know what the cost would be for your subwoofer being rebuilt versus just buying a new one. Here is one place that rebuilds woofers; there are several other places one can find also:
http://www.millersound.net/service.htm
