Silly question, I actually finished out my engineering degree as well (Mechanical), why electrical? You will miss out on a lot of really helpful classes such as Heat Transfer, fluid dynamics (related to flight), strength of materials, ect. I would personally look into becoming an AE, my buddy double majored at RPI for ME and AE, and just got an internship down in Dallas studying the corrosion on the skin for I believe for United or one of the companies. That gets his foot in the door, he can then continue training for his commercial ratings, ect.
Another route which would have been good as well is getting into a company such as General Dynamics, they pay for school, they also have links with Lockheed Martin...
Feel free to contact me with any questions with engineering! Im finishing up my last day of classes today in fact. Im in a similar situation with engineering, flight science degree, and ranking up hours. Currently going for my private license.