I have cleaned up what has delaminated. You think I should still delaminated the rest of the way?
Yes! I can't imagine that one limb will cleanly delaminate, while other is still perfectly bonded. Something has gone wrong during the glue up. No glue should cleanly delaminate if properly applied. If you fix the delaminated limb now, the other limb will quickly fail as well.
Yes, titebond 2 is fine. Make sure to remove all old glue and that both lams are perfectly mating.