People do move from QA to design positions, so it's not a crazy route to take to go EA. Although...do you know for sure what games you'll be working on? Because EA closed most of its console development--it's mostly Facebook games and The Sims now.
I'm not sure if moving from QA to design happens at Zynga--I do know it's hiring a mixture of 10-20 year industry veterans, and 40-year-old moms. Zynga's actually a very interesting place, though; they do things like have a game design question, and then just try both methods (with some "tiny" sample size like a million people) to see which is more effective. Zynga is kind of on the forefront of some very new game design (although not necessarily game design you want to work on, but damn, I'd love to be a fly on Zynga's wall once a week; I think there's a ton that the rest of the industry could learn from Zynga's experiments).
--
Cats land on their feet. Toast lands peanut butter side down. A cat with toast strapped to its back will hover above the ground in a state of quantum indecision