I'm not in the same situation, but I graduated as a mechanical engineer and did (do) game programming as a hobby. If you are not sure about a career and that you'd enjoy coding >40 hours per week then maybe mechanical engineering is a better choice, since it's easier to get various (easy or more demanding, creative or non creative etc) jobs in that field. And definitely easier than getting a game programming job.
Working as a mechanical engineer can also be a good base for some own little game programmer studio, since it's pretty good paying.
This path works pretty good for me, though I only started programming after starting college.