the car goes crazy with a bad alt but that's not the only problem in your case...check the voltage...mine shows more of a 14.6...down to 14 is fine too but anything lower is a bad/rippling alternator
you can fix it but it'll cost about half the price of a rebuilt one, IF you can find the parts, which I could not(few months ago)
I doubt that the acceleration problem is coming from the alt or the battery
get a few voltage samples and post 'em here...one off(discharged), one car on(jump it but be careful
), one after you shut it off...that explains a lot