In the past, owner give-ups have tended to be younger dogs, usually under a year old, who became too much for their owners. They had little or no training, may not even have been house-broken. A puppy for the kids that's no longer a puppy; the summer dog, not needed now school has started. The cute jumper that's grown enough to knock us over. It didn't take long for these young dogs to move into and and adjust to their new homes. With proper training, they became fantastic dogs.
Now, we see older dogs coming in. They are well-trained. Their owners love them and are happy with them. But the job is gone; the house is gone; the money is gone. Maybe the new place doesn't take animals. Often, there just isn't money to buy food for everyone. Sadly, with tears, the owner realizes the only real hope for their beloved pet is a new home. These dogs come into the system confused and upset. When will my family return for me?
Once re-homed, they seem to fear losing it again. (My Thor, after 6 changes in under three months, was depressed for a year.) My Sunna is working through what causes changes: 1st it was being near other people too long; then going into a new house; now, it's the handing over of the leash. After more than ten months, she is still not positive this is a forever home! I think trevort's dog is another example of this new rescue.
Much of their weird behavior is coping behavior. Thor would steal food, mostly bread, and hide it. He rarely ate what he stole and we never punished him for it. When the depression stopped, the food-stealing stopped. If these were children, we might think these were magical practices: Hide bread, stay here beside the bed -- do these things and I won't be sent away again.
Obedience class is excellent for these dogs. They shine. Their new owners shower them with praise. The class helps build their confidence in their new situation. Good instructors will notice if the owner is pushing the dog beyond its comfort limits or, equally bad, letting the dog get away with everything.
I think it takes these dogs longer to become comfortable members of the household. OTOH, their issues appear to be more psychological than related to obedience, which can be easier on the new owners. (The puppy tears up your couch, the "new rescue" hides food in it.)
That's my take, based on my tremendous experience of 2 of them!
kjd