My Woofer Sounds Like a Tweeter

Nah, just:

1) put on some music at a low volume & turn off the amp
2) disconnect the main speaker wire from the main speaker terminals
3) make sure the speaker wire tips don't touch!
4) turn on the amp (repeat step 3)
5) touch the speaker wire ends directly to woofer terminals inside the speaker (repeat step 3)
6) listen: bass or no bass (repeat step 3)
7) turn off the amp (repeat step 3)

See mom - no smoke!

3) make sure the speaker wire tips don't touch!
4) turn on the amp (repeat step 3)
5) touch the speaker wire ends directly to woofer terminals inside the speaker (repeat step 3)

Just an addendum to Dan's advice...

If you have a tube amp you may want to touch the speakers to the woofer terminals before you turn the amp on. It's not the best thing for a tube amp to be receiving a signal from your source with no load on the speaker wires.

Just use a couple wires and a battery(ie: AA, C, D, or 9V), briefly touching the woofer terminals. You can do the same test with a multimeter, set to "ohms". The speaker should either push out, or pull in, depending on the polarity of the hook up. If it speaker has ever been stored in an area with high humidity: there could be rust in the voicecoil gap. If it's ever been dropped: the polepiece or magnet could have shifted. Either would hold the voicecoil captive, and eliminate it's excursion. If the voicecoil is burned: you'll get limited excursion, and a lot of rattle(with music).