I graduated with a BS in Mech Eng, and a Masters in Industrial Eng. Typical progression that I've seen is most people start off aimed at aerospace (big paychecks), then realize its too theoretical or too hard. Then they switch to Mech Eng, which IMO is probably one of the widest fields. Those that can't hack Mech Eng usually go to Civil and from there switch down to Environmental or other less prominent fields of engineering. Chem E's and EE's (Electrical Eng) people tend to have been oriented in that direction from the beginning, never doubting what they wanted to get into. Though some decide Chem E is a little too dificult and switch over to Materials Eng (not much easier, lol).
In speaking with friends and collegues from various disciplines, we all pretty much agree that Mech and Electrical are about the best fields for the breadth of areas in which to work. Their principles apply to nearly every aspect of most businesses.
I'll speak primarily of ME right now, since that is what I am most familiar with and I'll correct a misconception - it isn't for just gearheads and other car enthusiasts - hell, I hate working on a car and won't even change my own oil. Lost of ME's are not automotive oriented, though the automotive industry relies heavily on MEs to function. From my ME background, I can get into part design, as you mentioned, but what kind of parts? Medical devices, satellite or Mars lander components, gas station pumps, clothing or equipment manufacturing machinery - nearly anything in the world that you put your hands on a ME probably had a hand in designing. But it isn't just about parts, let's take a step back and look at parts together - we also design systems - artificial heart systems, heating and airconditioning for any kind of application, hydraulic and control systems for production or mobile equipment (I personally worked on large mining equipment). We can choose to focus on the manufacturing aspects of things (leaning on Manufacturing or Industrial Eng skills) - how it is made, what's the best process, best design, best use of materials, etc. We can focus on control systems (which leans on Elec Eng skills), structural integrity of cmoponents or systems, production plant layouts and flows (leaning on Industrial Eng skills), etc. Other than Chem E and hardcore EE jobs, an ME has perhaps the best balance of background and skills to go into any situation and problem solve. We can stay a jack of all trades or focus on any particular aspect of ME that interests us for a career.
There is another question, what is your ultimate goal? Management? Research? Hardcore fields like EE, ChemE, and Materials Eng tend to be more research focused for careers. I can't speak for Civils, but Mech E's are offered opportunities to go into R&D, Management, or just kind of float in a production oriented career. I honestly feel the choice of ME provides the best opportunity to continue to evolve, change directions throughout your career, and still be competent and capable at what you do.
The most basic question, however, is "what interests you." Nearly any engineering field pays decently, and with any job you find a way to live on whatever you make. But the key to life is to find something you enjoy (I knew I'd be an engineer by age 12, lucky me). What interests you about EE and ChemE? I'd imagine during the first 2 yrs your curriculum would be fairly generic (aside from the required English, Arts, etc) as far as entry level eng courses that allow you to decide your major later. I'd use that first semester or two to wander the halls of the Chem E, EE, and Mech E buildings - find students studying in their Jr and Sr years and ask them what it is like studying in that field. You are still very young, and far from graduating, you may also want to hit the job fairs as they come through - talk to some of the recruiters about what their business does, what engineers they look for and what they expect those engineers to do - see what sounds challenging and interesting to you. The more input you get, the better informed decision you can make. Feel free to AIM, PM, or email me sometime if you want, I'm always willing to talk about any aspect of this stuff.