EXACT same experience. I bought the VZW Galaxy Nexus. Within a month I had 4 batteries as well. Not receiving text messages with low cell signal and WiFi enabled, couldn't get more than 4 hours of battery in LTE, so many radio issues. I swapped it out like 3 times just to be sure it wasn't a bad handset.

It's the first time in my life I've gone in early and bought a phone in cash off contract. I've not had an Android since.