I wouldn't let the drawing component deter you from one major or the other. The drawing involved in any engineering field is more about learning how to draw objects in different views (i.e. top, front, and side). There are classes that will help develop this spatial reasoning and teach you how to create engineering drawings. In my experience, I was required to take a class called Intro to Engineering Graphics where we learned descriptive geometry, orthographic projections, and how to draw multi-view engineering drawings. We then went on to a geometric modeling class where we learned to draw and design objects in a CAD program on a computer.
In my day to day activities at work, I need to be able to look at and understand engineering drawings and how the dimensions relate to one another. I occasionally need to draw a sketch to help someone understand a design concept I'm thinking of, but otherwise I don't draw anything. I think you should give both majors a try if you're interested in both. Maybe take some introductory classes in both and see which you enjoy more. Good luck!