Oh DEAR! You *will* have your hands full!
If you can make at least 2 of the carryons the 'roller' type, the five and four year olds should be able to pull those behind. Unfortunately the baby will need to come out of the stroller.
I posted elsewhere the other day that when my DD was little, I'd strap her carseat onto the stroller using a men's leather belt through the seatbelt loops of the carseat, attaching it to the back of the stroller. If you did that, you could then strap the baby into the car seat that's attached to the stroller, stash one bag underneath the stroller and hang one from the handle. Alternately, you could use a backpack for one of your carryons.
They may or may not make you take the carseat off the stroller, though, so you'd need to be prepared for that.
Do you absolutely need the four carryons? This might be one of those situations where more checked luggage is a good thing. Are the four and five year olds able to reliably hold hands? If so, have them hold hands walking in front of you.
Hmm... I'm trying to think this through logically. When you're in line for the security check, have the kids in front of you, each holding a bag (even if they're having to drag it). Once you're almost to the front, quickly unbuckle the baby and sit any other carryons in the stroller. Reach down with one hand and (hopefully!) unbuckle the belt. When it's your turn, with the baby on one hip, plop the bags up, then shoot the stroller over to the area where they 'pat it down'. If they make you take off the carseat, now's when they'll do it. If that's the case, plop it up on the conveyor belt as well. Only once everything is up on the conveyor belt, start letting the four and five year old go through. You go through last, carrying the munchkin.
Once through security, hubby better be there, or he's in serious trouble! If for whatever reason his flight is delayed or whatever, snag the stroller first and push it to the end. Unload your bags from the conveyor, stacking them directly around the stroller. Buckle carseat into stroller, buckle baby into carseat, divvy bags to the younger set and plop your other two bags wherever you'd been carrying them previously.
**Even though I'm not a fan of 'fanny packs', I'd strongly suggest carrying your money & valuables in some type of front pack on you. Since you'll be obviously distracted trying to keep up with everything, you'll be an ideal target for pickpockets.**
I'll never complain again about traveling with one!