I had my physical yesterday and received my customary yearly flu shot. My doctor (who is a woman) really tried to hard-sell me on the COVID booster. Even if I were inclined to get the booster, which I was not, I would not have done both shots at once. In 2017, I had the flu shot and a pneumonia shot at the same time (one in each arm) and came down with a 3-day case of the flu. A flu shot and COVID booster sounds like a recipe for disaster.
I only got the flu shots these past 6 or 7 years, because my Endocrinologist recommended it. I'm 74, and never had the flu prior to getting the shots. I won't be getting any more flu shots either, because you can't trust that the person doing the jabbing, won't inject you with the Covid vaccine.