Please look at the statement which I know is ok.
- I'm not going anywhere without you!
The second one I am not sure about is:
- I don't go anywhere without you!
Are they both correct?
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These are both "correct", after a fashion, but "I don't go anywhere without you" is an unusual phrasing that would only be appropriate in a specific context:
A: Do you understand? Are my instructions clear?
B: I understand -- I don't go anywhere without you.
We can assume that, previously, A told B, "Don't go anywhere without me", and so B is simply repeating A's instructions back to him.
Otherwise it would be more natural to say
I will not / won't go anywhere without you.
or, more formally:
I shall not / shan't go anywhere without you.