We employ a novel algorithm using a quasiexact embedded-cluster matching technique as minimization method within a genetic algorithm to reliably obtain numerically exact ground states of the Edwards-Anderson XY spin-glass model with bimodal coupling distribution for square lattices of up to 28×28 spins. Contrary to previous conjectures, the ground state of each disorder replica is nondegenerate up to a global O(2) rotation. The scaling of spin and chiral defect energies induced by applying several different sets of boundary conditions exhibits strong crossover effects. This suggests that previous calculations have yielded results far from the asymptotic regime. The novel algorithm and the aspect-ratio scaling technique consistently give θs=−0.308(30) and θc=−0.114(16) for the spin and chiral stiffness exponents, respectively.