L.M.
First, I would recommend that you cut back on the items you are taking with you. Most of the parks that you may visit in Orlando will have strollers for rental. The rental car companies have car seats you can rent. It is likely that any costs associated with rental will be cheaper than the cost to ship the items both ways. As I am sure you know, the airlines now charge for shipping items. Are you prepared to pay $25 for the first item and $40-50 for each of the next items (must pay for both flights going and coming). Some items you could purchase when you arrive in Orlando.
The fewer things that you try to carry on to the plane with you, the easier it will be for you to care for your twins and to get thru security checkpoint.
The ideal way to get to the airport would be to have someone drop you off - and this will save you the cost of parking your vehicle. You can either go to the lower level (designated on signs as "All Departures") where you can have Sky Caps check your luggage and items to be shipped there OR you can go to the upper level (designated on signs as "All Arrivals") where you can walk directly into the counters and check your luggage and items there. The difference is that you will have to pay an additional fee per bag if you check them downstairs (this is besides any amount that you may want to tip the Sky Caps).
If you decide to drive and park your own car, then I would suggest that you drive to the terminal and let one of you go in to check the luggage (either downstairs or upstairs again) while the other waits in the car with the children (one person can be dropped off right at the door and then the other can pull around to wait right there in the short term one hour parking). Once luggage is checked, you can all drive to the remote lots or one of the offsite parking facilities (I tend to use "The Parking Spot" because the buses are yellow with big black spots and they are easy to identify.....they have a website that you can check out. They are cheaper than airport parking and are nearly as close as the remote lots....either way you will have to take a shuttle bus.)
At that point, you will only have to worry about the twins and the carry-on items. Again, I would limit carry-on items to the things that you will really need during the trip (e.g. items to keep the twins occupied, food items - but no liquids - purse with personal items, etc.).
I think you will find that the trip begins much more pleasantly if you and your husband are not having to fuss with moving so much stuff, keeping track of it all while trying to keep track of the twins. There will be so many new sights and sounds and things that will attract the twins, that no matter how well-behaved they are, they may be tempted to wander off to investigate.
I wish you a safe and fun trip.