if it was me and wanted to spend time with the family on the 25th, leave at 11am or so, the 25th is not a heavy long distance travel day. go till about 7-8pm. get up the 26th and hit the road by 3am, kids will sleep the rest of the way in and you will get there around 10-10:30am a head of the traffic.
That is what I would do. I would plan to drive until you got as far south in GA as you could. When we left WDW the 26th last year we didn't leave until around 10 am and we didn't notice the traffic until we hit GA. So you left earlier in the morning from GA on the 26th you should be good!
I've never done a check in around 3am so I have no clue how that works. Maybe try calling the hotels and see how it works.
I know from my house in Cape May county they say it should take 13hrs to get to Brunswick, GA so if you hit the road at 11am depending on exactly where you are you should hit GA by 11pm. Its about 5 hours from Savannah to WDW so even if you get on the road by 6am I think you should be alright. Most people will leave the 26th so you already have a very good head start on them.
Since we are about 30 miles south of Chicago, right on the Indiana border, we will be taking the I65-I24-I75 route. We did this last March (2009) leaving early in the afternoon on Friday. We made it down to Louisville and may I say, never, never try to find a hotel room on a Friday night in Louisville in March. Talk about March Madness.
Saturday we got all the way to Lake City Florida, and then on Sunday we coasted into the Beach Club about 11am. We found that there werent many accommodation options between Valdosta and Lake City.
This year we are only taking 2 days to drive. We plan on leaving about 3am on a Saturday and stopping between Atlanta and Macon (probably McDonough), then leaving early on Sunday and getting to Caribbean Beach between 3 and 4pm. After reading the posts here, weve decided to make a reservation around McDonough for Saturday night (especially after our problems finding on room in Louisville). We dont really push while driving, but this way if we encounter any delays (weather, traffic, etc) we know we will have a place to stay that night. So thanks for the tip.

All in all, a great trip - I'd be happy to answer any questions if I can!