We write down the algebraic Bethe ansatz for string theory on AdS₃ × S³ × T⁴ and AdS₃ × S³ × K3 in its orbifold limits. We use it to determine the wave-functions of protected closed strings in these backgrounds and prove that their energies are protected to all orders in α'. We further apply the ABA to find the wave functions of protected states of AdS₃ × S³ × S³ × S¹ and its Z₂ orbifold. Our findings match with protected spectrum calculations from supergravity, Sym^N orbifolds and apply to the complete moduli space of these theories, excluding orbifold blow-up modes for which further analysis is necessary.