I have a similar problem. New iPhone 5S. Verizon service. Poor call quality (robotic sounds), barely able to hear people, voices fading in and out, dropped calls, poor call coverage in areas that seemed reasonably well covered with my iPhone 4. Upon reading a multitude of threads, articles, etc., I notice that folks with various other types of phones are having the same problems on Verizon. I notice that people with iPhone 5S are NOT having problems with calls on other networks such as AT&T. In my research, I learned that Verizon has a number of challenges with the 4G LTE network. One of the issues is that they have added so many new customers, and now bandwidth and congestion is a problem. They have a solution in the pipeline for next year to resolve some of this pressure. Other, more technical issues, are also implicated in the current issues with calls.
Although reception/voice processing hardware designed by Apple to accomodate the Verizon network confusion may be potentially be flawed, I'm relatively convinced that this is a network issue.
I have done all the recommended backflips and do not use a cover on the phone. (Turned off LTE, Bluetooth, reset the network settings, turned the phone off regularly, turned on data roaming, tried calls Verizon to Verizon, tried calls Verizon to other providers, talked to Verizon, talked to Apple, considered purchasing a flip phone....)