First off, I must say book through a travel agent, especially if you are not sure what cruise line or where you want to sail. I do group travel for a travel agency, and I know travel agents can steer you in the right direction.
I see many responders have advised you to check with Disney Cruise Line. They are great, nobody does it like Disney. But, they are also very expensive. This coming summer they are not offering a 7 day Caribbean sailing, on 3 and 4 day cruises.
Pros and cons of Carnival versus Royal Caribbean (RCCL).
Carnival Pro - Carnival's cabins are much larger on average than any other cruise line. This is talking about the staterooms, not the jr. suites or larger categories. So, if you are looking for the least expensive route, the least expensive cabin on Carnival is going to be a lot bigger than the least expensive cabin on RCCL. You can even fit all 5 of you in the cabin. Granted it's a bit tight, but it's doable and you're not in the cabin for a great amount of time anyway.
Carnival Con - Carnival does not offer adjoining cabins, RCCL does on a limited basis. So, if you want a separate cabin for the teens, on Carnival you would not be able to have an interior door between the cabins. If this does not matter to you but you want separate cabins, consider getting yourself an outside (or balcony) cabin and booking them in an inside cabin across the hall to save money. No matter the cruise line, you would have to book it with one adult in each room, then you can sleep how you want once on board.
Both cruise lines offer Kids Clubs for all of your children (RCCL they must be potty trained). Kids Clubs are great. And I practically guarantee that your teenagers will want to spend almost every waking hour there. Even the shyest kids love going there. Carnival also offers Teen Only shore excursions, they are supervised of course.
Carnival Pro- Carnival offers year round 4, 5, and 7 day cruises from Galveston. RCCL only offers 7 day cruises seasonally (Dec-Apr). From Midland you could drive to Galveston instead of the extra expense of flying somewhere.
Carnival is a lesser expensive cruise line than RCCL. They do cater more to the "fun loving" crowd. If you are going during the "out-of-school" months, you are going to find just as many families on both cruise lines. RCCL can be a bit more of a "refined" crowd.
I hope this bit of advice helps in your decision.