Go for the ME degree. AND take the EIT (engineering-in-training) exam!!! Get good experience for 4 years then take your respective Professional Engineer exam. This will open losts of doors for you. It will be tough road and you cannot go wrong. An engineer with "hands-on" experience is a hot commodity. Learn and practice working on teams. Get to know and understand technicians.
I am a Civil Engineer by education but work as a Patent Examiner at the US Patent and Trademark Office in the Electrical Engineering Arts, specifically examing dynamic information storage (disk drives, magnetic heads, etc).
An engineering degree will allow you to approach problems and issues in a systematic way. And with the hands-on experience, you can tackle the most problems. When designing, you will most likely understand what a technician will need or have to go through to provide maintenance. I know lots of engineers who do not know how to replace a simple fuse in their car. Don't be like that.
Good luck and study hard. hahaahaa