I just got the Galaxy Note 3 and I am noticing the same issue.
I have my phone in LTE/CDMA mode. When I got the phone it came in Global with roaming turned on. At first I thought that was the reason, but it continued to happen.
I think Verizon really needs to explain why areas blanketed in 4G seem to mysteriously drop to 3G.
I live in Avon, OH and we are solid 4G.