We left our oldest at that age for a cruise. SHe had a blast playing with Grandma and Grandpa and didn't even notice we were gone. My wife had a worse time of it than our daughter did. Don't worry about it.
And yes, they have since put Cellular at Sea (you can google on that company name for more info. But I have Verizon service and know the deal. For voice calls you will be charged the International Roaming Rate. I think you can buy a international roaming plan to make this cheaper if you need it but if you don't its $2.50 a minute. Its a quarter to send a text and 5 cents to receive a text. You should call Verizon and make sure they enable you account to use the service (add international roaming which I think they will do for free) We used texts for "emergency contact" and then used email from the internet cafe on the ship to send emails. If you HAVE to talk with the grandparents then the 2.50 a minute is still cheaper than the ship to land phone charges.
Hope that helps
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013