Fabulous question! Unfortunately, yes, you will not be able to become an electrical engineer if the college/university does not offer that degree program. However, that does not preclude you from taking classes at a local community college and then transferring later to a school that does offer such a degree. Specifically, you would take classes at the community college and transfer the credits to the college/university. This approach to obtaining a degree is becoming increasingly popular as the cost of higher education has increased. Many community colleges even have programs which guarantee acceptance at colleges/universities, if the student maintains a minimum GPA and achieves a minimum SAT or ACT score. If you decide to pursue this option, it is critical to consider the following: 1) the track record of the community college in placing students at universities, and 2) the courses which will be transferrable.
Regarding career possibilities, there are so many that I can’t do all of them justice. I’ll name a few to try to give you a sense of the diversity of your options. Medical robotics is an emerging field. This includes things like building robotic arms to enable less invasive surgery and faster healing times. Other field, more related to military and security applications, is the development of automated bomb detection and destruction robots. These can combine automated sensors for detection as well as robots for movement. Finally, artificial prosthetics has made enormous advances over the past 5 years. In just the past year, a robotic exoskeleton which enables people previously unable to walk was demonstrated. This skeleton leverages robotics technology. Therefore, EE, and robotics in particularly, truly has widespread impact.