Hey, fellow English/Classics student here! :-) In fact my first years learning Greek and Latin were in Boston. Been programming since I was 8, but always favored the humanities for formal schooling.

I like to say if you can read Greek you can certainly read Perl/C++.

If you're good I don't think you need the masters, unless you want to specialize in something mathy/difficult, e.g. machine learning. But if you can pick up just an algorithms and data structures course it will make you a better programmer.

Best of luck to you!