I think I would:
#1 check everything you can as opposed to carrying a bunch of stuff on the plane. ( it sounds like you are already doing that. ) That way you will have a minimum to lug through the airport on your way to retrive your luggage. That is your longest walk.
#2 unless you fear someone stealing your luggage, or you are in a big hurry, take your time on that long walk. We did this once, really by accident, and our luggage was against the wall waiting for us when we arrived at baggage claim, and the crowds were gone. We picked up our luggage and went to rent our reserved car. No crowds or lines.
#3 If it were me, I would as a group retrive the luggage, use one of those carts that are right nearby if you have to, and then as a group go to the Dollar rental area. One adult go to the rental counter, the other sit in the waiting area (right near car rentals) with the twins and the luggage.
#4 after getting your rental set, as a group, go to get your car. It really isn't that bad. You will probably get an elevator to the Dollar parking level. There are signs everywhere. ( i would think that you could bring that cart with you while the other adult holds the twins hands)
I would just rather stay together because at least it is known where everyone is and there is no chance of getting confused and separated, then you may start off your vacation crabby and mad.
I find it helpful to look at the web site when using an unfamiliar airport. Here's MCO's
http://www.state.fl.us/goaa