Airplane mode will certainly prevent you from getting charges, but the problem is it also disables Wifi and Bluetooth, which you don't want.
Turning off Cellular Data and Send as SMS, and not answering your phone if it rings, will take care of most of the problem. However you will still be able to receive text messages via SMS, which I think you still get charged for. If you don't get many text messages then this might be sufficient for you. I'm not aware of a setting to completely turn off the cellular connection, other than Airplane Mode which disables everything.
I have an AT&T phone so I can just pull the sim card, which completely takes care of the problem. I know Verizon phones for use only in the USA don't use sim cards, but I see that the Verizon iPhone supports GSM for international use. Does that mean it has a sim card? If so you could just pull it out to disable the cellular connection completely while out of the country. At least assuming Punta Cana is using GSM and not CDMA...