It's definitely odd that he's reacting to lamb. The only thing my golden retriever could eat without losing fur was lamb.
The suggestion for a food switch is usually to go with something that gets most of the protein from a single source so you can rule out other allergies. It's also suggested that you switch to a game meat, like elk, rabbit, pheasant, etc, because your dog probably hasn't encountered them enough to become allergic. But most foods with game as the main ingredient are pricey, and since he's having trouble with lamb, it's probably safe for you to try chicken.
Chicken is the most common problem food for dogs with allergies so keep a close eye on him after the switch.
I hope a food switch is enough to get him better! Keep me posted