See this post for a solution that worked for me. It is slightly different than yours because I could receive and not send.
I was also having the same issue. I received my iPhone 5 almost a week ago and after a few days I noticed that I couldn't send or recieve MMS from people that don't have an iPhone/iMessage. After doing my own list of trouble shooting and then contacting Verizon Tech Support and them basically repeating the trouble shooting steps I had already done myself, ultimately it lead to me doing a complete reset of my phone (Erase all Content and Settings). Now here is where I believe the problem is. I restored my iPhone 5 from a back up of my previous iPhone4 that was on the AT&T network which has different settings and which somehow changed the MMS setting for Verizon. I believe iMessage uses data and that is how picture messages were able to go through to other iphone users.
So basically I had to go into settings--->General---->Reset----->Erase all Content and Settings and then set it up as a new phone without storing from my previous back up which would put me back into same position. Again, this is because I restored a back up from my last iPhone that was set to AT&T settings and porting my number over didn't seem to be the issue at all.
The only issue now is to manually replace the content(pictures,apps ect) that I had previously.
just want to let you guys know of a better option then deleting everything you've got on your phone.
I went into: settings--->General---->Reset----->Reset all settings
Upon restart my phone was able to send mms messages.
Oh, and before this I had deleted all my old messages, b/c someone had the theory that if there was one undeliverable message it screwed up mms. That didnt work so my next step was the one above.
My phone was ported from straight talk (with whom i didnt have mms settings with either). I didnt have mms on my phone for like 3 months, finally i got around to looking into it again.
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