Forgive me if this was already something that you have tried, but it sounds like your phone is set on using internet only for connecting. If you haven't already done this, check your Wi-Fi Settings [Menu>Settings>Wireless & Networks>Wi-Fi Calling Settings>Connection Preferences] and see if you have your phone set on "Wi-Fi Only." If you do, tap the appropriate setting to take you off of that (i.e.) "Wi-Fi Preferred" or "Cellular Preferred").
Hope that this helps.