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Also, should I file as if I lived in CT (the state in which my university is) or NH (my home state)?
Yes, you would still be considered a full-time student for Education Credits if you were taking classes on a full-time basis.
Your home state is generally your resident state, and you would file there. If you had income from Connecticut, you might have to file two state returns, one for your resident state (NH) and one for your non-resident state (CT).
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