The first time we did the overnight drive I was very concerned. I kept telling my hubby that if we got tired we were stopping. I was worried we would be exhausted the next day. All the same concerns your guys are having.
Needless to say after that first overnight drive I won't go back to driving during the day unless necessary. Here is what we do.... About a week before our trip we make sure we don't stay up late. That means no staying up to watch tv or pack or anything. Forces us to get it all done early. We usually leave around 5pm and my hubby starts out driving. I let him drive as long as he wants to. That usually lasts until around 10 or 11pm. While he's driving I nap off and on. When he's had his fill of driving we stop to top off the gas tank, get the kids settled in their PJs (if they aren't already), use the potty, and stretch. Then I take over. The only rule I have is that if I am driving the overnight hours he is required to SLEEP! I put the volume of the speakers to the front driver's side and either have movies playing so I can hear them or music. If I need to stop, I stop. Everyone else is usually asleep and I actually LOVE this time "alone". As a mom of 3 that alone time is hard to come by. It's kind of like running, but without the physical part. HA HA HA!
Usually during this time we hit the beautiful mountains around Chattanooga, TN and the craziness of Atlanta, GA. The only different is that there is literally NO traffic. I love it!
Like I said, I stop if I need to. Usually the adrenaline kicks in and I realize, "Hey! I'm on my way to WDW!" There are lots of gas stations open 24 hours so if I need a cup of coffee or just need to stretch I stop. I don't push myself and we take our time if needed. I've never had any trouble being too tired.
I usually drive until the sun comes up. The kids wake up, we stop for gas, potty breaks, and breakfast. Then we switch again so I can sleep. My hubby usually drives the rest of the way or we may switch again after I've had some sleep. I'm kind of a control freak about "arriving" at Disney and love driving into the place myself.
We usually check in around 1 or 2pm, grab some lunch and our room is usually ready. Then we let the day take us where it wants to. If we are tired we nap and go to the parks at dinner. We may swim and then head to the parks later. Usually we don't head to the parks in the afternoon only because they are the busiest then. The first night is never a super late night so we can all get a good night's sleep.
Our kids do great in the car sleeping. It makes the whole trip go so much faster for them.