Who I have the same questioned this topic

I am currently traveling internationally in Europe and am receiving text messages with airplane mode enabled. I am not connected to wifi but messages are still coming through. I use the native Android Messages app and not the Verizon Messages+ app. I do not have calling over wifi enabled. I have a Galaxy S6 with Android v6.0.1. I am obviously concerned with the potential charges for this usage. Can someone from Verizon direct me on how to stop my phone from receiving text messages or confirm that I will not be charged for them?