I'm a bit alarmed to read of your Moderna experience. I am finally on the vaccine elegibility list and hoping for the Pfizer.
I don't know that alarm is really justified. A fraction of vaccine recipients get these responses, which are just the result of the immune system's reaction to the spike proteins produced using the mRNA from the vaccine — i.e. doing what it's supposed to do, if perhaps a bit more vigorously than necessary.
More detail on my experience: I got the shot on Wednesday afternoon, and felt fine that evening and night. I began to feel flu-y and exhausted by mid-day Thursday, and canceled a meeting and basically went back to bed. I certainly wouldn't have been able to play a sport on Thursday. I was beginning to feel better by Thursday night; by Friday, I was a bit bleary, but certainly well enough to go about my normal workday — I taught a class online, etc.; by Saturday, it was as if nothing had happened. My wife got the same shot at the same time as I did, and... nothing. Maybe a bit of soreness in her shoulder.
I had COVID a year ago — although it was too early for testing, it was confirmed by serology months later — and anecdotally people who had COVID are more likely to get these reactions to the first shot. Others are more likely to get them after the second.
I have not heard that Moderna or Pfizer or J&J are any more or less likely to produce side effects, so I don't know that you should bother optimizing on that score. My advice would be to pick an appointment time where next-day side effects, if any materialize for you, wouldn't be too disruptive to your life, rather than picking based on on which jab is on offer.
It's kinda ironic how life works out. As a liver transfer survivor I have a compromised immune system. I was eagerly awaiting my time frame to get the vaccine. Unfortunately due to a botched hip replaced surgery, I developed 2 viruses. My doctor has advised me not to get the vaccine until we are sure the virus has cleared my body. I do intend to get the vaccine when allowed to. I post this just to show that yes there are certain people who due to medical circumstances cannot get the vaccine. Also I post this because some people in here are nosey and would demand to know the reason why. Anyway go Sox!!
RGG's anecdote shows the importance of everyone who
can get the vaccine getting it when it becomes available, so that we can help confer the protections of herd immunity on those who cannot get it themselves for whatever reason.