Having a master's in engineering will probably do the two things you are looking at:
Help you get a decent industrial / national lab position. Personally, I'd try to get into a master's program where you can do a master's
with thesis option under the supervision of someone who is adjunct faculty and primary staff at a national lab. For example: when I got my master's in electro-optics (a joint program between EE and Physics), I worked at the Air Force Research Labs... I should have stayed there for a while before going to grad school in physics. The money and work was good.
But, especially if you haven't had prior research experience (I did, but I'm sure more didn't hurt), getting a master's with thesis option will give you a chance to get what (as a former member of a selection committee to a graduate program) the committee often deems most important: recommendation letters and a personal statement emphasizing research experience. In our committee, this factor was highly weighted in the scoring routine that we used to select new grad students to the program, although GRE's, grades, etc, factor in as well. Make sure your master's GPA is as good as you can have it (mine was a 4.0), and that the program is good as you can get into. And study for the GRE (my gre, was, I admit, pretty hideous, but I still managed, with my background, to get into one of the top-ranked programs in my field).
Having a similar background makes me perhaps more optimistic than others here. Of course, make sure that you aren't burned out of school before your PhD program. Expect it to still take at least 5 years... since you'll have to take the core classes (an EE master's program will probably only get you out of electives). I was a bit burned out by the time I got to grad school, so I didn't pick the best project (I picked the adviser with the Hawaiian shirt

). That isn't always best (publications out of our group were slim). Still, as a lecturer at the college level, and via the support of my department, starting some research program of my own, albeit relatively unfunded), I'm now doing what I'd like to do -- teaching (just not getting paid much, as educators rarely are)!