In this situation I'd want a professional evaluation of the dog by a veterinary behaviorist or a good trainer behaviorist perhaps one suggested by your vet. Petsmart does not come under that heading.
This is a young dog and may have issues due to mistakes made in socialization and training or he may be a dog with traits that limits his ability to behave peacefully with modern living conditions an stressors.
I find more dogs with training issues than with basic personality disorders. If the dog was over three years old I'd say you'd have a very difficult time changing behaviors but this age dog might be able to learn if given the right training.
On the other hand a known biter is a big insurance liability and a danger to your home and livelihood. I would suggest you view some of the videos here to see what I mean about changing behaviors http://www.veoh.com/collection/imotd
But if this is beyond your reach I do think euthanasia will be a better choice than rehoming the dog or leaving him at a shelter. The kind of people who want a biting dog usually are not the kind of people you would want to have your pet.
Hope this helps you!
If you had a vet behaviorist evaluate him with a poor prognosis then for your safety, your neighbor's safety and the dogs safety euthanasia may be the best choice.
I don't say that lightly. I've made this difficult decision with dogs that could not fit into today's society and I've also rescued dogs that were not treated well due to their willingness to bite and fight. Some dogs just have traits that might have been perfect for when dogs were needed to defend farm, livestock, family from intruders but that now need to be toned down from that to fit into what our current expectations are.
I will say though that it is absolutely normal for a pit bull to be animal aggressive but the human aggression is not correct breed traits at all.
I know this is an awful decision to make and I hope your vet will support you if that is the choice you determine on.
Dog Expert:Rescue, Train,Breed,Care
30+ yrs dog home vet care & nursing, rescue, behavior&training, responsible show breeding, genetics