I can speak with first-hand knowledge. I have worked in the aerospace engineering field as a structural design engineer and as a test engineer for the last 37 years. I've worked first for General Dynamics, then McDonnell Douglas (now part of Boeing), Lockheed Martin, Boeing again, Sikorsky Helicopters, Boeing Again, Northrop Grumman, and many others. Each in different cities and three different US States, none of which were my home state.
That being said; It was never ever boring. Quite the contrary. But Aerospace Engineers are always looking for the next project and/or the next company that actually has a new project just about every 2 to 3 years, maybe 4 to 5 years if the company has many projects.
The big aerospace companies have many divisions, albeit often in many different states and you could stay with the same company, but may still have to relocate. Los Angeles used to have the greatest number of different aerospace companies and I believe that may still be true. Seattle, Denver, and Huntsville Alabama are cities with several aerospace companies, so consider those towns if you want to stay in one place, but you will still likely change companies several times over a career.
In Aerospace the most "stable" work is in commercial aircraft. Next is defense work. The least stable but IMHO most interesting area of the Aerospace Industry is Space. It's not like automobile work, where a new model comes out every year, and manufacturing provides steady funds for researching the next big project.
Commercial aircraft orders from airlines tend to follow a 5 to 7-year cycle with LARGE peaks and valleys. This is driven by the retirement of aging aircraft, and to a lesser extent, fuel prices and engine technology advances that bring on the potential for better fuel economy. Defense and Space are mostly driven by government budgets. Space however is becoming more "commercially driven" but it remains to be seen if private space companies can be profitable without government contracts or at all.
Speaking on $pay$, most aerospace jobs are middle to high earning potential (for engineering) depending on where you live/work and what you focus on career-wise. AE jobs in California pay great, but the cost of living is ridiculous while AE jobs in GA pay O.K. but the cost of living is pretty fair.
The single biggest issue that I can foresee is a shrinking job market for this role. I may be wrong but my impression is that these jobs aren't that common. HOWEVER, you will learn a lot of knowledge and valuable skills in pursuing such a degree that will help you get other positions. AND if you have a passion for this area of study then you should definitely do it - there's no better work than that which you're interested in.