Is I am finished gramatically correct?

What is the difference between I am finished and I have finished?

Any help would be greatly appreciated.
I think old-time prescriptivists will tell you that "I am finished" will make you sould like a roast ready to come out of the oven.

In real life, no one will bat an eyelash if you say "I'm finished" (or "I'm done").
I disagree.

Correct usage is "I have finished."

"I am finished" means "I am done for. I am about to die."

And while I have said on several occasions, "I'm done," I have no illusion that this is correct usage--it's just colloquial.
You seem to be contrasting correct with colloquial. I think this is a false dichotomy. You can have incorrect formal writing as well as correct colloquial expressions.