gmail - How to switch a Google account without resetting a phone?

On my Android LG Optimus Dynamic phone I originally entered an incorrect email when setting up the Google account to use. I want to switch emails now, but when I add the correct account and then try to delete the first account, it says it is required by some applications, and that I can only delete it by factory resetting the phone. However, that message does not come up when I try to delete the second account, and it removes fine.

How do I delete that original account so that I only have the new one left without factory resetting my phone? (I don't want to lose all of my apps and my contacts).