ATT here too. They have a "passport plan" which gives you a much reduced rate for calls, unlimited texts, and a limited amount of internet for something like $60/mo (something like this, my knowledge of the details are hazy). If Verizon has something similar it is a good option. Call them, see the options and the cost for services if you don't get a plan.
In general - set your data count to 0 when you get on the plane so you can always see how much you have used overseas. Then turn off data usage for all apps you don't care about. You probably don't need your local news fetching content, using up data while you are away! Know that you can turn off all data usage (under settings - cellular) in general and just use wi-fi. That won't cost anything. You could then turn data on when you need something specific, like to look up a phone number or to get google map directions (last I looked routing isn't offline). Or just do your social media on wifi, which is easy to find.
Getting a different phone/service could make sense depending on the length of your trip. I personally like my phone and apps set up as I have them, and want people to be able to call/text me at my normal number so I don't get a new phone or SIM when traveling.