<begin tirade mode>
OK, there are a number of competing issues here. Start with your desires in life. Most people seem to design their careers based upon perceptions of what the big corporations and government offer. Phrases life "career path" come to mind. Also ideas like "eventually get into management" stuff like that. What you don't hear is anything approaching passion for the subject matter. It becomes the part of your life by which you make the money to do the stuff you really want to do. Which is fine, so far as it goes, but is also limiting.
Then you get the problems that beset the modern university engineering courses (including computer science) where there has been significant emphasis on dumbing down in order to keep enrolments up. This leads to some very short sighted choices.
Think about how long your career is likely to be. 40 plus years. In that time the technological pointy end is going to change out of sight. 40 years ago the fastest computer on the planet was a CDC 6600, which was hand crafted out of discrete transistors. Most universities only owned one computer, and electrical engineering was all about power distribution and conversion, electronic engineering all about RF design. Now, in your lifetime, where are things going to go?
The point? Get the fundamentals right. Which means getting the mathematics. This doesn't change. Look at audio. If you are interested in design for audio, especially the recording side, you need underpinnings. Much of what you need to know isn't taught as core curriculum subject matter. You won't find undergraduate stuff on ultra low distortion amplifier design for audio. But think about what is relevant. You need signals, if you don't know Fourier you can't reason about distortion, sampling, DSP, etc. If you want to design the next big thing in compressors you had better have a firm grasp of control theory. Even if only to understand what has gone before. In your career it won't matter what the latest and greatest technological marvel is, it will obey these fundamental mathematics and physics.
The danger I see now is that many students are leaving universities with poorer and poorer grounding in these fundamentals at the expense of sexy subjects that have transitory value. It is these graduates that see their jobs vanish offshore.
So, computing versus electronics. Well my background is computer science. Heck, I have a PhD in it, and taught CS at university level for some years. My main regret in course selection at undergraduate level. I didn't do the EE stuff. I am largely self taught. Now I work at an EE design tools house. But in software. But even in CS the fundamentals are being ignored. When I was an undergraduate numerical methods and numerical analysis were fringe subjects, regarded as not really CS, and largely disliked by the students (including me.) Now with the passage of time I can see the extraordinary folly of this. We have entire generations of graduates who write numeric code that is not credible. Issues you see in DAW code can come under this heading too. Ironically I almost ended up teaching the numerical methods course a few years ago. Partly because I was one of the very few who was passionate enough about its importance. Again, no matter what the changes in technology, the underpinning mathematics don't change.
So, go the EE route. Pick the core subjects, see what really grabs your passion. Don't just think in terms of a nice easy career path. If you commoditise yourself, you will be treated as a commodity. Big government and big corporations regularly treat their employees as commodities. Outsourcing your job is always on the cards. Middle mangers are always the first to go in a crunch (sales guys are the last sadly) and a 50 year old unemployed middle manger usually never finds another job in the field. But if you have the passion to pursue a particular area there is vastly more about. Somehow I can't imagine Eveanna outsourcing Manley's design team to India.
<end tirade mode>