ThriKreen said:
Do the software engineering. Game design can involve a lot of logic design and number crunching - the more you understand how a computer works from programming, the better you can design your game systems based on what computers can and can't do (or shouldn't) do.
Same applies to software engineering. Game programming is more complicated, but depending on the software you're making, that can get complicated, too.
I am a software engineer (did just start though), and I can say unless you love programming, do not go into it. You can end up working long hours (coworker worked 79 hours last week), and it is very frustrating. You say you have a lot of experience with programming? What exactly have you done? Asking because I was one of the better programmers in my classes, and then when I got into the field it seemed like I knew next to nothing.
With languages, from my experience in looking, most business are looking for C# or JAVA (or legacy languages such as Ruby on Rails, COBOL, etc.). I know there are some places that use Python and C++, but not nearly as many. I have little experience with C++ and Python, but I can tell you trying to learn C++ was a bit more than just syntax changes. It is object oriented, but you have to do a ton more with it that other high level languages take care of for you.
As far as game design goes, they have people that are focused on programming, art, sound, etc. So you probably wouldn't have to worry about having to do something you aren't good with.