Sounds a lot like an issue that I recently experienced on an '89 Firebird Formula that I own: I suspect your starter is shorted out. It happens, and the symptoms were a lot like what you're describing.
Unfortunately, the only way (I'm told) to verify this is to remove the starter and bench-test it. And if it is the starter, be sure to load test the battery after installing a new starter; I had to replace both on my Formula, as the shorted out starter apparently took the battery down with it...
X3 w/6-speed & Sport Package:
A sports sedan disguised as an SUV...