We consider a model of a polymer in Z^{d+1}, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched non-negative random environment. Alternatively, the model describes crossing random walks in a random potential (see Chapter 5 of [Sznitman] for the original Brownian motion formulation). It was recently shown, by Flury and by Zygouras, that, in such a setting, the quenched and annealed free energies coincide in the limit N to infinity, when d is at least 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.