How often will you have the suitcases in the car? I'm on my phone so can't see your ticker if you have one, but consider this if the only time all of you and all your luggage will be in the car is to and from the airport:
Keep the car you have, because a full size car isn't much more than the smaller ones, the big price jump is between the full size car and the SUV or minivan. One adult signs up for Magical Express and you tag both suitcases to be transported by Disney. That adult rides Magical Express alone. Youll be ready for a break from each other by then

The other adult takes kids, car seats (That hopefully still snap in to stroller), and stroller, gets car, and drives (luggage-less except for the diaper bag, which stays with the kids) to the hotel. That adult starts checking in when they get there. Just make sure.whoever has the kids know how to install the carseats.
If your car seats no longer snap into the stroller, get two "Prince Leonhart Car Seat Check Bags", which you will use for years to come to carry the car seats even if you don't check them.
The other option would be to have both adults go to the rental car, get it loaded, USING the front passenger seat for luggage, then one adult taxis/Uber to the hotel alone from the rental car place.
And yes, I have twins. My girls are five now, and we've done permutations of this for years. Another tip: when you get to your hometown airport, do something similar unless you're being dropped off. One adult drops the second adult, all the luggage, stroller, and car seats at the airport terminal (or even where they let you check bags from the drop off lane) then drives the car to long term parking and takes the shuttle to the terminal to meet everyone.
Especially with twins you have to divide and conquer. Have fun!