OK, I finally had some success, but not from my trip to the Verizon store. They were no help at all, so I took my own actions to try and fix it. (About 3 weeks ago, sorry for the delay in notifying everyone...) I completely uninstalled all of my Email accounts, including the Gmail account, for 48 hours. I then re-installed them from scratch, and lo and behold, all the accounts now update normally EVERY DAY. I've changed the sync schedule around several times, and it seems to follow my instructions just fine. I have no idea as to why this worked, but it did.
This may be hard for some people, especially if you use your phone for work, but at least it fixed the problem.
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