Downloading ringtones shouldn't make any differences although ringback tones may impact this as they are not compatible with VZW's Advanced Calling 1.0. Advanced Calling 1.0 on the iPhone with VZW enables voice and data at the same time. Make sure this is toggled on:
Settings > Cellular > Enable LTE > ensure Voice & Data is enabled. With this toggled on you will be able to do voice and data at the same time. Let us know if this resolves your issue.
Ok, it sounds like you added Ringback Tones either by accident or what not. Ringback tones are not compatible, yet, with Advanced Calling 1.0. I'd remove Ringback Tones online and then try to add it back.
I removed my ringback tones when Advanced Calling 1.0 launched as voice/data at the same time was more important to me and the calls to other VZW Advanced Calling 1.0 customers is more clear in terms of audio.
It's wasn't my ringtones it was my ring back that people hear when they call me. The Verizon rep remove that I rebooted my phone it showed voice & data I now have ringtones can talk & surf the net at the same time. The rep had to call me back since this is my only phone she was great