It sounds like Fleet 1580 might have what you're looking for. We are a fleet of mature (30+) officers who all have jobs, families, and other responsibilities but play STO for fun. We are a smaller fleet, and that has made for a very friendly gaming experience for all of us. Each of the fleet's officers know the others, and so there's lots of friendly banter and helpful advice on the Vent channel. We highly value teamwork, and our veteran officers help out newer officers to learn the game. A couple of us were in closed beta, and most participated in open -- we've had a blast.
We are located in the Eastern and Central US timezones. Since we have jobs and wives, and many of us have children, fleet events are never mandatory and there is no minimum weekly time commitment.
We do use some light RP elements, such as fleet ranks and awards, but we're not, properly speaking, an RP fleet. Each member does as little or as much RP as he wishes.
We would be honored if you would consider joining us. We could certainly use a Lt. Cmdr. engineer, and we have several fleet officers at about the same level as you are. Our fleet site is here:
http://fleet1580.guildlaunch.com. If you have other questions, please feel free to PM me. I wish you the best of luck in finding a fleet, and we hope to see you in-game!