I am currently working on a Computer Engineering degree. My school offers a game emphasis for this major, and I am going to take it. I discussed it with my adviser and mentioned that I probably would not want to actually program games for a living, but I want to learn how so I can try to write small games on my own. He told me to go ahead, if I don't go into games professionally, no one will ever care about the emphasis area. I am leaning toward trying to develop new game controllers. My school is doing quite a bit of research in that - if you have heard of Blind Hero (guitar hero controller for the blind), it was made at school. I am hoping to find something that I really enjoy doing. I am finishing up my core and math classes this summer, so it should be all fun classes from here on; Computer Science, Computer Engineering, and Electrical Engineering. My fall schedule seems like it will be a blast, albeit a ton of work. I am taking my 3rd c++ class, and this one will hopefully teach me enough to actually write something interesting. I am taking a microcontroller class, so I will learn a little 8051 and ARM assembly, and we will also be using and FPGA. on the EE side, I get to take my first circuits class. Probably my least favorite will be a networking class, but I hoping I will find it interesting because it is probably one of the better areas for a job. To round out my schedule, I have my first Game Dev class.
I switched to Computer Engineering last semester after discovering that I had no interest whatsoever in Civil Engineering. Try to find a career that you are interested in! I was in Civil because I thought it would be easier to find a job, but I was dreading school, let alone the next 30 years of work. Now that I am where I think I belong on the computer side, I am really looking forward to school and my future career. It is a lot harder, and sometimes I get the feeling that my brain is slowing a bit (I will be 40 almost 40 when I graduate in a couple of years), but I am really enjoying the challenge. Except for calculus, I struggled through all three semesters

. I think calculus is pretty easy to understand, but the algebra I learned and loved in the 80s has largely been forgotten. That made it really rough to actually get to the calculus part.
I am definitely listening to all of the people warning of the time commitment in Game Dev, but I am keeping the option open. I am really leaning toward the hardware side, but I will know more by the end of the next semester since I will finally get into the topics deep enough to decide where my strengths and weaknesses are.
If you decide to go to school, which I would recommend - it may not be absolutely necessary, but it surely cannot hurt - do not expect to be an expert when you finish. You will definitely be entry level and will still have a ton to learn. There are just too many topics to cover, and it looks like all you can hope to get is an intro to each during a 4 year degree. I have taken 2 programming classes so far, and received As in both, but I still do not really know how to write a program. I can write the simple stuff we did in class, but as soon as I try to do something that would actually be useful in the real world, I realize how little I know. This is where the coders will start off with an advantage over the college grads, there is no substitute for experience. I think I am learning the fundamentals better though - at least I better be! Two semesters, and they will still not let us use the string or template libraries.
Good luck! Find something you enjoy. I go to school with quite a few people who have given up 6 figure incomes to try to find something they enjoy doing. You are going to spend a huge portion of your life on your career, as someone who has wasted a couple of decades doing jobs that I hated, I can say with certainty that it is not worth it.