Unfortunately, many companies in industry get Computer Engineering and Computer Science mixed up, so be careful if you chose CE. Not that one is better than the other, but most HR departments will see you as some sort of programmer if you're CE. Starting out in industry, the only jobs I found available as a CE were more along the lines of VLSI, signal processing, and hardware infrastructure (think rooms of servers that serve as processor farms). Computer Engineering is so much more than that, but to get to the fun/hard/good stuff like processing architecture design and PCB/Circuit design, you have to either go pretty far in your education (MS/PhD) or have lots of experience from the right job.
As an EE, I found I was able to take any course from the Computer Engineering side as well as any course from the Electrical side that I wanted. I took embedded systems, Microcontroller Design (used a PIC18F452! with assembly!), advanced control systems, a Sensors and Perceptions course, and an electrical mechanics course. These are a fairly broad range of courses and I apply most of what I learned in them to the prototype and build seasons in some way.
My career path right now is as a Systems Engineer, and most of what I do has had to be learned on the job. During the FIRST season, I have fairly sound inputs to the mechanical side of things on the robot even though I'm most knowledgable about Electronics, Programming, and Controls. The point is, the reason I love mentoring with FIRST is that it allows me to do the engineering I love to do from multiple areas of engineering but am not able to do every day at my job. It is indeed the only way I can do everything I want to as an engineer currently, but eventually I will become project manager and be able to touch multiple disciplines in my job as well.
Hence, just going through engineering classes will change the way you think enough that if you're patient, eventually you can do everything you want anyways. There are very few careers that deal with everything a FIRST robot deals with within one job (it's always delegated out among several employees), but eventually you'll be able to work on everything. In the meantime you can mentor a FIRST team
edit - Note that the jobs I found for CompE's listed above were for the Atlanta area. Not sure what else is out there for entry level engineers in other parts of the country.