Verizon used to have great customer service. Now it is just as bad as Sprint. I just made my third attempt to deal with technical support team. After waiting 1.5 hrs to get someone on the line he hangs up on me. It is more likely that the call was dropped because that is what happens when I try making calls anywhere outside the city, but he didn't even bother to call me back. In fact, he took it very personal when I said there were known issues with the Note 3. He was more concerned about telling me these are the best phones on the market. Well, if that is the best Verizon has to offer then I am better off with Sprint. The issue was not resolved. Every time I call each tech has a different story about what is wrong with my phone. This last guy tells me to keep my phone on roaming at all times so I can get a signal. Seriously? That is not going to fix the poor reception or dropped calls. It just stops the roaming message from appearing. What about the text messages that I can't make? What about the 3g signals at 4g prices? Thankfully I am no longer under contract. I can take my 250 bucks I pay every month to Sprint for unlimited 3g.
Alma, I'm not sure that Verizon can do much for you in your situation. You need to do a little research and contact Samsung. Samsung either needs to provide you a different model phone, or potentially a Note3 with a new build date that doesn't have the problems (people aren't sure what exactly is wrong here).
Generally, your device enabled the Band4 capabilities of the phone's radio (this is a new spectrum of frequencies for cell phone use) with the KitKat update (abbreviated: KK).
Prior to the update to KK, it was using Band13 frequencies. If you wanted to, you could go to Wikipedia to read up on this Band4 and Band13 stuff.
Band4 signals are known to penetrate buildings differently than Band13. Regardless of phone and model, if it works with Band4, I think it has to do some special work to hold a good signal with a tower. Your device is having a problem doing that indoors, and you are not alone.
There may be something wrong with the low-level radio or radio software in the device. It could be the antenna or some signal amplifier that isn't handling Band4 as well as other devices that were XLTE-ready and got the KK update.
Note3 band4 radio problems
Note3 band4 service menu