I'm about to start my first year of med school and I graduated for a top 5 engineering school. Mostly cause of the in state tuition though, not so much the "quality" education. First i would like to dispel some myths.
1-You won't have time for everything-Going into engineering in any school with a tough program means you are a very motivated individual and likely know the work load you will be encountering. Most of the engineers I am friends with are as busy as I am, granted with other things. Working for a consulting group, being on student government, playing NCAA sports are all things my close friends do and we go out as often as any other engineer and far more than many of the pre-med kids. You will, of course, be very busy, but there is no reason why you can't make time for everything.
2-It will be tough taking extra classes-The addition of biology and extra chemistry is difficult but even in the most rigorous engineering programs you are expected to have 20-30 hours of non-engineering classes. Most students entering into competitive engineering program have most of this completed with AP credit. This just means rather than taking the blow off classes like some of your friends you'll be taking more difficult classes. Though, with proper planning there is no reason why you can't graduate in 4 years. I finished all my pre-med requirements by the end of my sophomore year and took the mcats before the start of my junior year. There were one or two really difficult semesters but it really was not that bad.
3-Your GPA will hurt you-Its true that some schools will weigh engineering into your GPA. However, if it is like my school where our engineering program is much much much more difficult than the biology program, where a 3.3 in engineering is much harder than a 4.0 in bio, you are going to be screwed by the system. They are not compensated for to that degree. This is the bitter truth, though i find no reason why anyone driven enough to take on engineering and medicine can't pull off a half way decent GPA.
now for the advantages. I feel that these FAR outweigh the extra classes.
1-You take AWESOME classes-I'm a bioengineering major and I LOVE every single class I take. I feel that being a straight biology major, I would enjoy it just as much but I would be shortchanging myself and miss out on some of the incredible classes I have been in. I would have never discovered my love for quantum mechanics or heat transfer.
2-It sets you apart-Honestly being able to say I spent two summer as an intern at an engineering company where I was in charge of whatever it is you did, makes a huge difference. Often in engineering companies you need to work with people with some crazy personalities and those are all skills you can use to your advantage to show that you can, in fact, interact with even the quirkiest patients.
3-Its a backup plan-I entered with an incoming class of 23 students. Of those, 15 wanted to go to med school. We had 8 people follow through and actually apply. 7 people got in. The one student that didn't sent in his applications over winter break. So it was more due to him being lazy than anything else. Every other person was thankful they had a job or grad school opportunity they were interested in rather than being part of the large percentage of biology students who wanted to go to med school but now had no idea what to do.
My biggest advice however is plan plan plan. You need to know what you need to do and when. I have friends who were chemical and electrical engineers who both made it into med school but both decided their sophomore year. They both graduated in four years but had a very grueling last two years of college. You know now that this is what you want to do. Figure out what you need and when to take your classes. Get help early. Become friends with engineering students that want to go to med school as well, they are an invaluable resource, pre-med advisors are useless to engineering students, I have been told incorrect information on multiple occasions by multiple advisers. You will need to make sacrifices, but it won't be any worse than and biology student.
BTW I concur with the previous poster, fluid dynamics and materials classes are insanely hard and extremely math intensive. I do not envy you for the years of partial derivatives you have ahead of you.