  We present a Gaudin-like determinant expression for overlaps of $q$-raised
N\'eel states with Bethe states of the spin-1/2 XXZ chain in the non-zero
magnetization sector. The former is constructed by applying global $U_q(sl_2)$
spin raising operators to the N\'eel state, the ground state of the
antiferromagnetic Ising chain. The formulas presented are derived from
recently-obtained results for the overlap of the N\'eel state with XXZ Bethe
states. The determinants as well as their prefactors can be evaluated in the
scaling limit of the XXZ spin chain to the Lieb-Liniger Bose gas. Within this
limit a $q$-raised N\'eel state that contains finitely many down spins
corresponds to the ground state of free bosons. This allows for a rigorous
proof of the BEC Lieb-Liniger overlap formula for an arbitrary number of
particles.
