sounds like your starter could be bad...take it off and have your local auto parts store test it. i just had the same problem on my 03 taurus.
i also have a problem with my cavalier...1988, it runs great, just had a problem cutting out (not missing) just a quick burst of stutters. then yesterday it just stuttered then stopped...now she turns over...and over....and over...not even stumbling to a start. anyone know where the ignition module is on this thing?