'Költözöm' is the correct, but a lot of people would say 'költözök', so don't worry about that. I don't know, that you've ever heard of '-ik verbs' but maybe Wiki can help you: https://en.wikipedia.org/wiki/Hungarian_verbs#-ik_verbs
It's a bit long, but I'm afraid, I couldn't explain it properly.
I think "költözök" "is" the "correct" one because it's an -ik word, isn't? "költözöm" would imply an object. Anyway: Both are fine. There are verbs that are supposed to end in -m but but people say it with -k because the -m version sounds pretentious and there are also verbs that should end in -k but people say it with -m because it sounds nicer, softer. The problem is grammar books are written by gramma nazis who refuse to follow the natural change of the language.