9 month old pup itching!!

I have a 9 month old lab/terrier mix (I got him at 6 weeks old from a lady that had a litter and didn’t know anything about them) (Also I know 6 weeks old is too young, again she didn’t know what she was doing & I would rather him come to me than a bad home).

He is my big baby!! He’s very active and so funny! He loves his toys and treats!! We have taught him sit, lay down, speak, shake, and high five! But he has a lot of hot spots and sores on his back right above his tail, on his tail, and all on his back of his back legs. He had missing hair all over there too. We recently switched him to a better brand of hard food, thinking that what’s causing it, but he’s been on it for about 3 weeks now and there’s no change. I bought Burt’s Bees spray for itching and that seems to help a little but he’s still itching. I’ve given him one bath a week with dawn dish soap since it started. Nothing it helping and I’m not sure what’s going on!! He doesn’t have fleas or ticks. I would take him to the vet, but we really don’t have the money right now. I feel sooo bad, and I want my baby to be comfortable again!! I felt so bad I had to go get him some more treats and a new toy and new collar!! i

His new collar, he was so excited!!❤️

This is before the itching started!!

This is the best picture I could get of the spots that he is scratching. You can tell where the bald spots are. My poor baby!!