To test, please try to send an MMS using your Android stock messaging application (WiFi ON and Mobile Data OFF) and report back your findings? When doing this test check the 3G/4G data indicator does not come on during sending (indicating that your phone is turning mobile data back on).
Note: Verizon does not support MMS over WiFi. You must have Mobile Data ON to send / receive MMS. For users running firmware versions prior to Lollipop Textra can do this automatically for you (if you normally leave your data off), see Textra Settings > MMS Settings > Activate Mobile Data. For Lollipop firmware and above this setting is not present as the functionality required to do this has been removed by Google, so it's not possible.