Most ethical vets will no longer snip the vocal cords - it's considered to be inhumane by most. And it does not solve the problem which is behavioral for the dog.
I would want to talk with the vet about medication to help with separation anxiety for the dog, to help during the transition. I would also leave the radio or a tv playing. There's even a homeopathic anxiety medication called homeopet (www.homeopet.com) that we have had a lot of success with.
As far as shock collars go, they are only as good as the person working the controls. Instead, your mom might want to try a citronella collar that will spray a blast of smell in the dog's face when he barks. It's much more humane.
But I would not even try that since I really think this is because the dog feels insecure after the move and just needs time to settle in and perhaps meds to help.
Let me know if you want to discuss this more.