Since I have a smoker, hopefully someone will come by who does do smoking on a grill - I know it can be done.
I just found this for you for another
C@H pulled pork method, it doesnt' have the root beer, but the method you can use in a gas grill. Good luck and be sure to come back and tell us how it turned out.
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Combine all rub seasonings as shown below.
Coat the shoulder with the rub, massaging it into the meat. Place chunks of wood in a foil pack—use two layers. Bundle packet leaving a hole at the top so smoke can escape.
Coat shoulder with rub. No need to let it sit—it’ll have plenty of time to absorb flavors on the grill. Preheat one side of grill to med.-high. Place smoke packet over the fire, drip pan under unlit side. Place pork over drip pan and grill for 3 hours. Check every hour to make sure there’s plenty of smoke. Smoking takes 3 hours. Add wood packets as needed to keep plenty of smoke circulating. Once smoking is complete, wrap pork in two layers of foil, crimping edges to seal in moisture. Return to grill; cook 2 hours. After 2 hours, insert thermometer into thickest part of meat. You want to hit 180°. Keep foil around the meat and let rest off heat for 30 minutes. It’s now ready for pulling with fork or fingers.
(this below had pictures with it but here's the info anyway)
Because smoking takes 3 hours and you want to keep smoke circulating, make sure you have extra packets of wood. Add more as needed.
Don’t worry about this red ring around the meat. It’s not undercooked but rather a natural reaction to the smoke.