I am experienced at flying with 2 small children. Here is my strong advice:
Let DME handle your bags. But pack your carry-ons with your needs for the first 24 hours of your trip. If your checked bags run late for some reason, it won't really matter, because you'll already have with you everything you'll need for that night and the next morning.
Even if DME didn't exist, I'd still pack my carry-ons that why (and I do when we fly other places). WHY? We do it just in case the AIRLINE delays our bags.
DME will handle/deliver bags for flights landing as late as 10 p.m.
The last time we flew to WDW was before DME started the "get your own bags if you land after 10 p.m." rule. We landed right around 10 p.m. We had well-stocked carry-ons. My plan was to ask my resort's front desk to hold our DME bags 'til morning. But in all the excitement, I forgot. At 1:16 a.m., our phone rang ... it was Bell Services asking if we wanted our bags now or in the monring. I chose morning, and went back to bed. At around 6:30 a.m., I called Bell Services and requested our bags, and they were at our room 5 minutes later.
And by the way, about DME luggage timing ...
Disney says to expect your bags within 3 hours of resort check-in. SO in my case above, we landed around 10 p.m., we checked-in at POFQ around 11 p.m., so getting a call that our luggage was ready at 1:16 a.m. means that it was ready for us 2-and-a-quarter hours after check-in, which is within Disney's promised 3-hour time frame.
Believe me, with 2 young kids, the last thing you want to do is be at the mercy of the airlines and hoping they don't lose your luggage ... pack for your immediate needs in your carry-ons. And remember, each ticketed passenger is entitled to one carry-on and one personal item, even if they are too small to carry them themselves. My advice is let DME take care of your checked bags.