I had similar symptoms that also improved greatly with a gluten free diet.
But regarding my problems with GERD, it seems that gluten or no gluten did not make much of a difference and the only thing that helped was to follow the diet recommended in the book of Dr Jamie Koufman.
Basically avoiding coffee, tea, chocolate, seltzer, tomatoes, spices, garlic, onion, milk products, fat (particularly animal fat), citrus, and all foods with Ph level below 5.
I eat chicken breast, lentils, beans, lean fish, rice, potatoes, most vegetables, bananas, melon, watermelon, oatmeal, salt, water Ph 8.8 between meals, and no more than a few nuts and a tea spoon of olive oil per meal.