Charlie, to what internal temperature are you cooking the chicken? A few years ago the FDA revised the doneness guidelines and now 165F is considered safe for both the breast and thigh.
I like to put a salt-based rub or sometimes some herb butter between the skin and the meat and I do this several hours before cooking. Then grill indirect at 350-375 until done and if I want a really crisp skin I’ll put the chicken directly over the coals for a couple of minutes.
“Roadside chicken” is a popular method these days and a web search will get you lots of results. Also, since you own a Weber grill the TVWBB website might be of interest to you.