Is the battery light on at any time? Have you had the alternator checked? Any new Audio equipment installed recently?
The fact that it runs with the battery cables off tells me the Alternator is probably doing it's job. If nothing has been added to the car that could cause battery drain when not running such as audio equipment and this just happened out of the blue I would suspect the battery cable and terminals for corrosion or damage. These car's wont run well when the charging system has issues and DC voltage is particular.
The best way to figure out what is going on would be to use a DVOM and but it on the battery when you know the battery is full, say after a charge and start the car and monitor the voltage readings. I know my car stays around the 14.4 to high 13 volt range. If when your monitoring the readings the voltage goes down and does not come back up them you have a issue as described above.
Hope this helps.
