In terms of chemical shift, if you are comparing similar environments you'd expect the protons next to the azide to be further upfield (lower shift).
Without knowing a bit more about the rest of the structure it's difficult to answer the question. It would be useful to know:
For RCH2NH2 and RCH2N3
What is R? At least show all the protons that could be coupling to your CH2s if you are not in a position to divulge the entire structures.
Try to work out the pattern of the "multiplet", perhaps it's a dd, ddd, dt, etc. or even two overlapping shifts (check HMQC or HSQC spectrum).
Can you see the NH peak? Does it couple to the multiplet or the triplet in the COSY spectrum?