I do not believe that mms works over wifi. When I recieve a picture message on my s4 even with wifi turned on my cellular data kicks in and when I recieve the message, then goes back to wifi. If you have to actually turn off wifi in order to recieve an mms then I would think the hand off from wifi to cellular data is experiencing issues. (beginning of my post)
MMS is a system that uses the phone network to deliver messages to other devices. The routing/Delivery is done by the phone system itself, and so cannot work over wifi.
If you're concerned about data usage/charges on your phone connection, and want to use your wifi connection instead.. don't send MMS, send an email.
MMS requires at least a third generation (3G) network to enable such large messages to be deliverd.
I read your post thoughly. You wrote However, the only way this phone will send or receive MMS now is if Wi-Fi is turned OFF, which means MMS now runs my data plan usage up, which it is not supposed to do when connected to Wi-Fi.