Here's an update to my story, hope it helps. After being denied a new phone in my first go round with Apple, the the same issues were occurring after the restore, as I said. Yesterday, I lost LTE service early in the day and watched as the phone sat and sat and sat and did not acquire it back. Checked twitter to make sure verizon wasn't down, it wasn't. This time, I decided to NOT toggle airplane mode or reset the network settings so I could show Apple (since the first genius didn't believe me) that even when there's LTE available, the phone just isn't seeing it. Made another genius appointment for that evening. Ran errands at lunch which required me to go to three different areas of Manhattan - upper west, flatiron, and midtown. And all during that time, not once did the phone get LTE service again on it's own. Not ONCE. This continued through the evening and into my appointment with Apple. Showed the genius and he swapped out the phone for me quite easily this time. Even in their own store which I'm sure has a network extender (and where just 2 days ago at my first appointment I had had 5 full bars of LTE) there was no LTE service on the phone.
The 2nd phone is working MUCH better. Service in my office is still lackluster (in terms of bars), but the difference is the phone is connecting to LTE MUCH faster and STAYING on it, not losing it or hopping back and forth. In my office elevator, where I was never able to connect previously, the phone hung onto the LTE service most of the ride and I was able to load twitter and use iMessage no problem. Never happened before. And when walking around outside, the phone picked up LTE right away after exiting the elevator, versus before when it would require an airplane mode toggle to turn back on.
Point of that longwinded story for my fellow NYers/anyone: I'm sure part of the crappy service in certain locations is Verizon's fault, however, I wouldn't go switching carriers before getting your hardware swapped at least once. I'm much happier with the new phone. It's finally behaving how it should. Give it a shot!