Re: battery or alternator?
If the lights are dimming bad, you have a bad battery or bad conection. If you have an internal shorted battery, even a jumper/charger will not work.
Does your battery charger allow you to start the car while charging? Does it say so? If not, you will eventually blow your charger.
Here is one thing you can do to diagnose the battery at home. Disconnect one or two of the terminals. Directly hook up pos and neg jumper cables from one car to the disconnected battery cables on your Crossfire. This way you are eliminating your Crossfire battery completely. Be sure you have good connection and they are not touching anything to short out. If the car starts, you have a bad battery.
James