I go to UC San Diego, and they might do things differently here, but let me give you my perspective.
I am majoring in CS, going for a BS degree (they also offer a BA degree in CS, but I don't think many people do that), and am in the second quarter of my third year (3 quarters per year), so I've taken a LOT of courses for my major so far. I have a good friend who was a roommate of mine freshman year, and he is a CE major. We took the EXACT same lower division courses, and so far the vast majority of our upper division required courses have been the same. The only difference so far is that CS majors have a choice between a few lower division courses to take to fulfill a certain requirement, which includes a Physics course on relativity and quantum mechanics, while CE majors must take that course. I happened to choose to take that physics course, but I could have chosen to take a Bio or Chem course as well.
Note that I have, as a requirement, taken courses such as computer architecture, where we created a (simple, yet functional) CPU at pretty much the logic-gate level. It also includes a ton of programming courses, mostly C, C++, Java, and some Assembly. As electives I've taken courses on Databases (which included SQL and schema design), and am currently taking an elective in server-side web applications using JSP.
Looking at graduation requirments for CS and CE for my university (
http://www.ucsd.edu/catalog/CSE.html), it appears that the only real difference is that CS has more required electives (you can choose what to take out of a huge number of courses), while CE specifies a requirement for specific ECE courses, such as linear systems, electrical circuits and systems, and a few others.
So, if you were going to my university , switching to CE really wouldn't help you, since you'd have to take the same courses as you would if you were a CS major. In fact, CS gives you more choice for what to take.
But, alas, every university is different, so I would look very closely at the graduation requirements of a major before you decide to switch to it.