'I've got', 'you've got', 'we've got', 'have got'.
Technically, 'got' is a past participle, so such usage is correct.
It's just impure and lazy.
I have gotten sick and tired of such idiocy, and so I have to yell at you.
Think about 'have got' for a while. Doesn't it seem strange? If 'have got' is correct, why do we have the word 'gotten'? I prefer to use 'got' only as a simple past.
Avoid 'have got'. So long as you're using pre-made structures of this sort you'll never free yourself from common language usage.
Remember the Sapir-Whorf Hypothesis: language shapes thought.
Blindly using preset forms like 'have got' and dead metaphors such as 'toe the line' sets you into certain grooves of thought. doubleplusungood.