Pontiac Grand Prix

I have a 91 grand prix gtp with about 320,000 km on it and it just dont seem to want to work right. I figured out that the problem is every so often, only after its turned on it shifts into the highest gear and runs like that the whole time. Sometimes it kicks out. Sometimes it doesnt. When I go it slowly gets up to speed seeing as how it is stuck in a tall gear but I can take it on the highway and it will not shift. It just stays in the same gear and goes about as fast as it can. Even when I shift it into 1st or second(this is an auto by the way) it still does it. But I think I actually have to shift out of those. If anyone has an idea about this let me know please.
July 16, 2007.

Start at the ignition switch, then the neutral safety switch, starter relay, then starter.

Load test the battery to check its condition and double check terminal connections.

Or have someone bang on the starter while attempting to start-doing so and it turns over-sounds like starter.

Jul 16, 2007.
Won't hurt to check the alternator output, perhaps also the battery is not taking it could be sulfated.

Last battery change?

Jul 16, 2007.