Does "wanna" have a modal function here? Yes, it does. I think of it as being like 'You ought to . . . ' You should/wanna see a doctor about that cough.
You'd normally write it in full as 'want to', unless you were really trying to write dialogue that captures pronunciation, like 'ya' 'for 'you', etc.
Having a modalish function doesn't make it a modal though. You can't put it in sequence with other modals (like "he can and should go") and it has a past tense.
It is a conciously abnormal item, a conscious "perversion" of the general "want to", and therefore not worth classifying, since unless something goes nicely into a system there is no point giving it a grammatical name.
<<It is a conciously abnormal item, a conscious "perversion" of the general "want to", and therefore not worth classifying, since unless something goes nicely into a system there is no point giving it a grammatical name.>>