I suspect most, if not all, testing sites only offer one vaccine type because it simplifies training and reduces medical errors. When training thousands of people, you keep it simple - one dosing regiment, one needle type, one storage requirement, etc. Also people don't end up with one dose of Pfizer and one dose of Modena by accident.
I got my first dose of Pfizer on Thursday morning at one of Georgia's mass site and was impressed by fast and smooth the operation was. It took just under 25 minutes from start to finish with most of that time waiting 15 minutes at the last station for a reaction.
For symptoms, the upper, outer arm muscle, where the injection occur, got sore a couple of hours after injection and lasted about a day. The soreness was limited to just that muscle and it felt like I overworked that muscle.