Well, it so happens that I am an Air Force engineer-- Department of the Air Force (DAF) that is.
There are two kinds of Air Force engineers. DAF and USAF (Active Duty).
If you want to stay a civilian, go DAF. DAF engineers are civil servants (just like NASA or the IRS), and work on USAF projects. Project types range from commodity purchases (how to and what to buy to keep airplanes flying), to repairs (airplanes get hit by trucks or ground fire and need to be fixed), to reverse engineering (a 30 year old design needs to be updated to keep it flying), to basic science (at Air Force Research Labs), to safety investigations, to policy setting. Yes, most of your big projects will end up with some type of contracting--but if you are of the right mind (if you like the big picture, want to make sure dozens of major assemblies all work together, want to work hand-in-hand with pilots and gunners on aircraft, catch on to wide varieties of new designs quickly) then this is also fun. You generally won't be stuck on one actuator of one aircraft for months, and you generally will have a broad range of experiences looking at lots of parts of the airplane. No, you will generally not become a CATIA or FLAGRO expert. Other than that the only down side to being a civil servant aerospace engineers is that it is not a particularly mobile career--you can move between USAF centers across the country, but it is difficult; however, know that the Chief of Staff of the Air Force is trying to change that mobility problem.
Active Duty USAF aerospace engineers do much the same as civil servants, but they move around much more than the civil servants do. Yes, after about 8 years, a typical military engineering career will start to take on more program management. But there are also opportunities within the military to be flight test engineers (which has got to be the world's most fun job), space launch operators, and even to career broaden into maintenance or operations. Active Duty personnel can retire after 20 years (vs 30 for Civil Service), they can deploy to forward locations and fix aircraft supporting ongoing operations, and they wear uniforms--but that is about all of the difference between an active duty and a civil service Air Force engineer.
More to the point is the mission. You have never felt as good about your job as when you have shared a beer with a pilot of an aircraft that made a real difference in the Global War on Terror, and when he starts to list and personally compliment you on the many modifications and repairs that he knew you engineered. Or when he shows you pictures of his crew using your systems in close support of US soldiers under fire. Or when you and your team rush design a modification that top people in DC highlight on TV as a success story. You can't do it for the pay or to rise up the corporate ladder, but if the USAF Core Values (Integrity, Service Before Self, and Excellence in all we Do) define you, this is it. Even on my most frustrated days I have known a level of job satisfaction far beyond many of my peers.
Good luck in your decision making.