Common in small breeds. Both of our dogs have very sensitive throats. Make sure you walk the dog with a harness/leash, not the collar/leash.
Weight loss should help - less fat constricting the problem area. A food I've had good luck with is Wellness Core Reduced Fat. Make sure you feed the amount for what the dog SHOULD weigh, rather than what she weighs now. IMO, the amount recommendation listed on the bag is too much. I fed 1/2 c. daily (divided into two 1/4 c. feedings) for a dog that was supposed to weigh 14 lb., so your dog should need less than that - maybe 1/4-1/3 c. daily? I supplemented with about a tablespoon of no-salt green beans (hunger works wonders if your dog is picky) in the evening meal.