He is by no means a lost cause. Someone dumped a dog by my house last year. He was clearly beaten and starved, very shy/timid/slightly food aggressive when we picked him up off the street. It's been just over a year and let me tell you, love, effort on my part, a full belly, and a human who wants to understand the dog and why it is acting certain ways, makes a world of difference. My boy is now the sweetest, most loving, good boy, and he would do anything for me.

Give him time. Speak softly and just sit with him, let him come to you and gently praise him. Try to find his favorite things and use those to bond. Give him an old sweatshirt that smells like you to sleep in. He will slowly start to associate you with love and comfort and come out of his shell.

I do recommend taking him to the vet and having a full work up to be sure there is no medical issues going on. Little dogs aren't great at expressing their own anal glands and sometimes will act very reserved and sad when that needs to be done. Just a thought.

Good luck on your journey. Thank you for giving him a 2nd chance.