There are multiple models of iPhones these days but only to function on the different LTE bands. If you've got a 6 or newer, you'll be fine to connect overseas with a Verizon model phone. How your carrier bills is another thing. You can turn off international data roaming in the settings so you don't get some random huge bill. If your devices are paid off, ATT should allow you to unlock them and move to a different carrier. Also new iPhones are going to be announced on the 7th, so don't buy new just yet.
I had T-Mobile in Norfolk and was perfectly happy, but they were terrible in the Lower Alabama/West Florida area for service so we switched to Verizon. Three lines for $90ish after the mil discount. As others have said, it's not cheap, but it works well.