The latest COVID shot I got definitely made my arm sore for the next couple of days. That was back in July so maybe it's not the same one?
A lot of times, I have a sore arm from the flu vaccine but this year I didn't, just the tired feeling others are having.
I'd say get them both on the same day, but make sure you don't have to be anywhere for a day or tow-just in case.
I once made the mistake of getting a tetanus shot and a shingles shot on the same day, though, so just about anything else seems mild to me.