Do I need a Real ID if I have a passport?

I'm flying for the first time in a couple years- Chicago to Boston, so domestic. I have a drivers license, but it's not a Real ID, and in the last year, I've also gotten a passport for the first time.

I know the passport is sufficient to avoid the fee the TSA now charges, but I was thinking of getting a Real ID so that it will serve as my drivers license, and save me from having to carry my passport as well, since this is just a domestic trip.

I could go to the Real ID Supercenter and wait in line, but I was thinking of making an appointment at one of the full-service DMVs instead. When I try to make the appointment, it asks if I already have a passport, and because I said yes, it won't give me the option to make an appointment.

Are they just reserving appointments for people who don't have another option? Should I lie and say I don't have a passport?