My left arm is paralyzed so I ALWAYS get my shot in that arm. Since it never moves, it’s never sore!!! I sometimes have NO after effects or occasionally I’ll run a low grade fever for a day or 2 afterward. I’ve been taking them for 30 years & the ONLY shot that “made me sick” was my very FIRST one. My doctor at the time recommended that I take 1 Aleve before going in for the shot & take another once I return home. That seems to help control the fever. The occasions where I’ve had a low grade fever were the ones where I neglected to take the 2 Aleves as recommended.
The other thing that I seem to remember is that they say that it can take up to 2 weeks for the shot to become effective & anything within those 2 weeks, you would have had whether or not you took the shot. Hence, your coughing & sneezing have NOTHING to do with the shot!!! The queasy might be psychosomatic because you expected to become sick.