Have you tried clearing the cache partition? It's unlikely to help a lot but can't hurt, and it's harmless.
Also, does this happen when you are on CDMA only most of the time, or LTE?
If CDMA, perhaps try disabling any apps that may be interfering with the incoming call (caller ID apps, that sort of thing) if you have them? Caller ID apps often do a quick data lookup to find the name, and if you are on a signal that does not allow voice and data at the same time (like Verizon 3G), the use of the radio for data may be interfering with its ability to connect the voice call properly.
If LTE, maybe the phone is unhappy trying to use some of the Enhanced LTE services and Voice-over-LTE - try disabling those features - it will not interfere with using LTE data at all, it will just force your calls to the older and perhaps more reliable CDMA network.
As far as getting someone to help with the problem (assuming none of the above works), since it is receipt of phone calls that is the basic problem your best bet is to start with Verizon. However, no matter who you contact, their first suggestion will probably be a factory reset.