What you describe is very common. The mRNA vaccines were Phase 3 tested with three week and four week intervals, so that's how they are being delivered. Not to say that other intervals would not work fine, as long as they were at least three weeks. In the UK, I think they are giving Pfizer 2nd shots at 12 weeks just to get more folks partial protection quicker. Mine was at 5 weeks because they ran out of Moderna locally from the Big Freeze in Texas.
Nobody "feels it". Everybody has to wait 15 minutes, and gets a little card. Don't lose it: It's good for a year of free donuts at Krispy Kreme!