I don't have time to sharpen the axe, ...
I haven't got time to sharpen the axe, ...
I've got no time to sharpen the axe, ...
... I've got to chop wood.
... I have to chop wood.
... I need to chop wood.
... I need to be chopping wood. (= right now)
... I've got to be chopping wood.
... I have to be chopping wood.
All those sound equally okay to me, though in AE at least, 'don't have' is the most common. In more formal writing, you would need a semicolon instead of a comma, because there are two independent clauses, but that would be unusual in the written version of a spoken sentence like this one.
The only things that sound odd to me are 'the time,' which usually means the time of day, as in, Excuse me, do you have the time?, and 'as,' which is much more formal, as if in German you were to say '... denn ich muss Holz hacken.'
If you use 'as,' however (which I don't recommend), you must put a comma before it.