Firstly, a game isn't all about programming, it also requires art (2D sprites, 3D models, animations, etc.), music (sound loops, sound effects, etc.), and design (gameplay, GUIs, etc.). So just because you can program, doesn't mean you can make a good game.
Making games doesn't happen overnight, even with an engine. Using a pre-built engine would be an easier option, but it may not be so easy to learn (since it's easier to understand your own code), and also you probably wont learn as much as you would from developing a game from ground up. In the end, it is up to you on what tools you wish to make. I'm not saying that you shouldn't use an engine, you could use an engine, then eventually move away from engine entirely and produce your own code (to learn a bit more of what is happening under the hood).
If you want to have an actual career in game development/programming, I suggest you start with small, simple games and then work your way up to what you want to do (baby steps). I also suggest that you do continue to learn C++, since it will more than likely help you get a job in the industry, as a programmer. Although, if you're going to Uni, you're more than likely going to learn it, but teaching yourself C++ in your own time will do no harm.
EDIT:
I forgot to mention, that restricting yourself to one language isn't wise. Once you master C++ and you are comfortable with it (you've made various programs and applications), feel free to move on to other languages and learn about them, such as: Java, C#, Python, etc. This will generally make you a better programmer; learning other languages doesn't mean you'll forget a specific language (it's kinda like riding a bike, you just dont forget).
Also, with an engine or without one, programming knowledge is still required (and useful even if the engine doesn't expose programming entirely). If you really want to use C++ for games, learn it gradually until you feel comfortable enough to make games.
You could use this to guide you on what programming skills you require to make a game, on your own:
http://moosader.com/resources/tutorials/game-programming-prerequisite-test/1
If you're serious about this, then I wish you good luck, it's a rough process.
