No, please don't give up your dog. I read on a few other forums where someone gave up 2 dogs, 2 cats, ferrets and what not. It was heartbreaking the way they got these pets on a whim and disposed of them like they were garbage you put at the curb.
There's always a way to keep your dog. Even if you have to buy a cheaper brand food for a while, do without a few sodas, any other frills until you get back on your feet. Some food banks are carrying pet food now, you could use them until your luck changes. Even put off the shots if you have to or borrow from a relative who would be happy to help I bet.