Hi daisey, congratulations on your daughter and up-coming cruise! I think you've made a great decision to leave the little one with her grandparents. Maybe in a couple more years, when she's able to enjoy more of the ship's ammenities-Camp Carnival, the kiddie pool, etc., y'all can take her with you. She'll be hooked immediately
Check with Verizon before you leave. We had service in many ports, but they charge a roaming fee. I believe they have packages you can purchase for international calling that would be less expensive than your cell phone, but you could check first. When we were at sea, we did get some signal depending on where we were (like just leaving a port or sailing near islands). Just be sure to check first, so you won't be surprised with a huge bill.
They also have internet packages you can purchase, so that might be an option, too. Satellite service is improved, but can still be spotty. I know the ship to shore radio/phone always works, but is expensive. All ships have to have better access because more and more they rely on on-line services.
We have AT&T, we had service in St. Thomas, San Juan, Antigua, Tortolla, St. Johns, Virgin Gorda, Aruba, Bonaire, Cozumel, St. Kitts, Barbados, Grenada, Grand Cayman, Jamaica, Grand Turk, Nassau, Freeport, and a few others that I'm forgetting. But...most of them had roaming fees.
Hope this helps a little.