On your return flight home, if one of the carriers offered, you can check in at the hotel. (for a fee--we did this in Hawaii and it was well worth the fee!)
The only other time you need to handle the luggage is checking in from your home airport.
We use the strollers as our "carts" in the airport. We have made our 4yo walk and the 2yo would ride or I would carry. I would use my free hand for a carryon. My 4yo would use her hands to tote her little bag on wheels. If your children will be all over the place....then into the stroller they go. Then hubby gets 2 bags piggy backed and you get too bags piggy backed. Carryons I assume are diaper and activity bags--those can go in the bottom of the stroller and the extra can be piggy backed with the others.
If umbrella strollers, we have hooked the bags to the stroller with the kid in the stroller or one other bag in the stroller as a counterweight.
We also have straps that allows our luggage to be fastened together and a suitcase can ride piggy back with another.
Worst case scenario, if you don't want to purchase new luggage, you can do curbside checkin if you like and just pay the tip.
Honestly--once you get the schematics down, you will be amazed at how well you can get around when it seems you can't. We also have bags for our car seats (purchased at One Step Ahead) and they have straps and we can carry those on our shoulder.
The straps to attach the handles of bags together I think can be purchased separately. Ours has it built into the bag and the link we just store in the zipper pocket.
Another consideration--pack less....
Statistically, most often, travelers pack way way way more than what they need to a fault. You could probably trim down your packing as well as pack more compactly. Roll clothes instead of folding them for example.
Our most challenging trip was overseas to Europe. We managed--but the challenge wasn't getting into and out of airports, it was managing with trains and cabs once we were there as well as the long walks from baggage claim to trains and trains to airline checkin. That was with a 3yo and an 18 month old.
Depending on the trip, the girls share a big if it is full sized, or their stuff goes in with dad and my things. Their "suitcase" is used for carryon and has blanket, change of clothes (if early flight we drive them to the airport in their jammies) or jammies (to change into for a late flight) and a toy.