I'm having a very similar problem with my Verizon Galaxy Note 3. I got mine in October and it has been great, none of the voice quality issues that others have had.
About a week ago my phone started going in and out of roaming at home. The 1x signal averages around -82dbm and the LTE signal is around -104. The LTE signal is stable, but the 1x signal will suddenly jump to -100 or more and I will have no connection and it will roam and then eventually come back. I can't make a call or send a text during this time, even though it shows a decent signal and roam. I never have problems anywhere else. My signal seems rock solid at work and everywhere I go. My wife's LG G2 doesn't have near as good of a 1x signal (-88dbm to -96 dbm). Her phone never roams, never has call quality issues.
I have had 5 calls with Verizon and 1 with Samsung this week. Verizon re-provisioned my line which made me think it was fixed as I didn't roam for more than 5 hours then it started roaming and went back to as it was. I finally did a hard reset today at Samsung and Verizon's suggestion. No luck at all, just a major pain. I will see if they will replace my sim card tomorrow. I don't know where to go from there. My phone works great everywhere but home. Verizon did check my area and said the towers are working fine. One caveat, this started not long after the latest update in late December to fix connectivity issues. If it was that update, I figured the hard reset would work, but no luck.
Any ideas? I would love your help troubleshooting,
Jeff
Murfreesboro, TN
Verizon Galaxy Note 3