Whatsapp is an app that is used all over the world - except in the US. (People in other countries are genuinely shocked that almost no one in the US had heard of it - they use it ALL THE TIME). It can be used for texts, calls, or video calls. If you use it on Wi-Fi, there's no charge. If you are not on Wi-Fi, it uses data. If, in turn, you have a lot of data (such as on your home unlimited plan, or from a European SIM), you can still use it pretty freely. But if you don't have a lot of data, or if you're paying for data by the megabyte (such as with Mint Mobile), this "free" app can get very expensive, and/or can use all your data allotment. The big advantages of Whatsapp are that people reach you the same way when you are traveling as when you are home, and that anyone you want to contact locally (such as a hotel) is likely to use it. The disadvantage is that anyone you want to contact on it has to have the app installed. When you do install it, you'll see everyone in your contact list who has it already. You will probably be surprised at the number of people this is (I sure was).
If you have a choice of domestic plans, and don't have a reason to favor one over the other, then choosing T-Mobile or Google Fi makes sense, since they are clearly the best US plans for foreign travel that don't involve getting another SIM card. But Phred, be careful; they have tried to make sure that people use them primarily in the US, and only occasionally abroad. They reserve the right to cut off service if they determine that you are using them mainly or only abroad. How they define this is not publicly disclosed, but they do say that you need to have used the phone in the US for a certain amount of time before using it abroad. Plus, it's pretty pricey to get a plan that requires monthly payments that you will only use a few times a year. So, I wouldn't look to get T-Mobile with the idea of using it only when traveling.
So, of the things you listed, here's what I'd say:
1) just use my current Verizon phone and turn on an international plan?
If the cost of this in total (regular plan plus extra $10 on days you use the phone abroad) is workable for you, that's fine. Do remember to put the phone in airplane mode on days you don't want to be charged. I've heard too many stories of people who didn't think they had done something that counts as use, but it did, and they got charged without wanting to be.
2) Or get a T Mobile account in the States and only use it on the trip?
Again, don't do this, but if T-Mobile works for you in the US, it's a great travel plan, as long as you only need slow data. I have T-Mobile, and find that the data works for maps (slowly) and loading web pages like this one, without much graphics or video (again, slowly). It isn't really workable for other things, which I do on Wi-Fi.
3) Or buy a burner phone and get a local SIM knowing that calling home will be expensive?
You don't have to buy a burner phone; you can just get an unlocked phone, and use it in the US and abroad. And as I said, don't be so sure that calls home will be expensive. In addition to using the SIM itself for calls, you can also use Whatsapp (to people who also have it), or Google Voice (should work to any US number). I don't know all the ins and outs of Google Voice, but it seems that it's replacing Google Hangouts Dialer, which allowed free unlimited calls any US number (including 800 numbers) while on Wi-Fi. There was even a way to receive calls. Our poster Andrew H had posts with all the details of Hangouts, so I hope he'll update us on Google Voice.
Whew! That's all I have the energy to type now.