Hi Bev, Yes you will definitely be charged for data usage, you should turn data off on your device in the phones settings menu.
I'm guessing you have an Android device which you can find the data on off toggle in settings under data usage, just uncheck it to turn it off
Regardless of whether you buy a post-paid roaming plan or not, the phone will automatically connect in the US for voice, text and data, unless you manually turn these off or remove the SIM.
Buying an add-on only shifts your cost from pay-per-use to whatever is provided for in your bundle. Like Paul says, disable the data. There's no point in getting app update$ or $yncing file$ over the air when all that could be done for free when you're in range of wifi.