  We develop a new method to compute the exact overlaps between integrable
boundary states and on-shell Bethe states for integrable spin chains. Our
method is based on the coordinate Bethe Ansatz and does not rely on the
"rotation trick" of the corresponding lattice model. It leads to a rigorous
proof of the factorized overlap formulas in a number of cases, some of which
were inaccessible to earlier methods. As concrete examples, we consider the
compact XXX and XXZ Heisenberg spin chains, and the non-compact
$SL(2,\mathbb{R})$ spin chain.
