That happens w/ me when my stutter gets intense but it feels as if my lungs forgot to breathe normally. Not an SLP but I'm pretty sure this could be not-anxiety related, keep in mind people stutter even w/ those they are comfortable with. The tightness in your throat is just a side effect/symptom of your stutter. It could also be worsened (not caused by) anxiety. Going to speech therapy might help you a bit with that, even if it's just for a few sessions.