If I was landing in Tulsa or Minneapolis or Las Vegas around 10:00 p.m. I'd make sure to have my PJs and toiletries and clothes for the next morning packed in a carry-on bag with me just in case the AIRLINE delayed my luggage. I do no differently with WDW and DME. I can't imagine flying unprepared.
Go into using DME assuming your luggage will take 5 or 6 hours after you land (it really shouldn't take that long!), pack in your carry-ons accordingly, and you'll have a great time.
Packing for 24 hours takes up surprisingly little space. Check it out for yourself. I do it anytime I fly ANYWHERE.
I left for WDW on May 8, the 4th day DME was in operation. We were prepared for the worst. My luggage "magically appeared" in my room about 4.5 hours after our plane landed, about 3 hours after we checked in (MIL-related delay at the airport, 53 minutes from DME counter to CBR, and then time in line waiting to check in). When my luggage arrived, my wife and I were in the Trinidad North pool with our 2 pre-schoolers enjoying the start of our vacation. We were able to do this because we were ready for the luggage lag. For each of the four of us, we packed PJs, toiletries, "beach shoes," swimsuits, change of clothes for night time (just in case of spills, etc.) and a complete change of clothes for the next day. We also brought in our carry-ons a nebulizer and Xopenex, as well as diapers (for our 2 year old), snacks, and the kids' favorite bedtime buddies. We don't have huge carry-ons, we just pack efficiently and, like I said, it truly doesn't take up as much room as it sounds like it would.
ENJOY YOUR TRIP!