Just a few things I have discovered going through similar things with my girl.
Food allergies/sensitivities/intolerance are a pain in the butt to figure out. Mostly because it can take just one meal/treat to trigger a reaction and weeks for it to clear the dogs system.
Raw diet is frequently recommended and I don't disagree that it is easier to manage food issues feeding raw. The problem lies in the way most start raw diet feeding. Chicken is the easiest protein to come by in the US and is the cheapest protein as well for most. However, chicken is frequently the culprit in the allergy scenario. The majority of kibbles are chicken based as well. Thus why whether we feed kibble or raw we don't see any improvement.
In my personal case, that was part of the problem. My girl can't have chicken (or any poultry for that matter). It only took letting her have 1 raw pork rib with meat to discover pork was a total no-no, too. She had a full body rash and relentless itching in just a few hours after consumption. It took a week of dosing with Benadryl before we saw improvement in symptoms.
You have to be diligent in checking ingredients in everything your dog consumes. Especially treats. I turn down offers of treats all the time when we are out for this reason.
Without testing, it is impossible to know if it is a combo problem with environmental allergies as well. It's winter but where I am we don't have much snow. We still have some itching, but it's minor.
Before you know it, you can spend a fortune playing the trial and error game, like I did. And, you dog is miserable all the while. If I had to do it over again I would have just allergy tested in the beginning and saved myself and my dog the misery.
I have heard great things about Dr. Dodd's Nutriscan food intolerance testing. I personally want to do the test to be certain what my dog can and can't have to eat. As soon as I have the money (which I'd have if I hadn't tried so many different dog foods and fed chicken)I will do the test.
I currently feed raw. Beef, venison and goat seem to be safe so far as I can tell.
If you and your vet suspect allergies I personally would just spend the money upfront and test, if you can afford it. Know what your dealing with and go forward.
This has just been my experience and I hope it helps you.
Best of luck!