I think you make a solid point about the weather in that part of the country. It is a gamble and may be worth driving for that reason alone.
We just got back from our drive to Disney from Massachusetts. It's also 19-20 hours of drive time from here. With stops and bathroom breaks it is obviously longer, but we did it in two days- choosing to stop for the night both ways. We drove last year, too. We have stopped in Fredericksburg, VA, Fayetteville NC and have friends who have stopped in Florence SC. All of these places are good, safe options with direct access to I-95 and reasonably priced, clean hotels. I only recommend stopping in VA on the way home. It makes day 2 a very long day if you don't get through some of the Carolina's on day 1.
Last year we rented a car on points, but this year we drove our own minivan. The rental is the way to go, for sure!!! Too stressful taking own car, and I explain more below. Get a minivan or SUV w/ 3rd row. If you think you will be dealing with bad weather in your area the SUV is probably the way to go. Although we serviced our car before departure, the check engine light come on in NJ and we had to stop at a Toyota dealership to have it looked at. It ended up being nothing to worry about, but it set us back 3 hours and we hit horrible traffic. We had planned our departure time around avoiding high traffic areas at peak times, and that unexpected stop really blew our itinerary to shreds. If we had a rental, it probably would not have happened. We also left the weekend before the eclipse and that made traffic on that Saturday so horrendous. Even before the check engine light came on, we hit bumper to bumper traffic in NYC at 6am on a Saturday. With the unplanned dealership stop, we then sat in heavy traffic from NJ to Southern VA. It added a substantial amount of drive time to our day 1, and we did not reach Fayetteville NC until 9:30pm. We had originally planned to arrive in Florence SC, a few hours south of Fayetteville, by 4pm. Day 2 was a breeze. We left NC at 8:30am and checked into our hotel in Orlando just around dinner time. The kids were amazing through all the unexpected bumps in the road, and really handled it well. We were just grateful that there was nothing more serious and we were still able to drive our car with no major repairs necessary and arrive in Disney at the time we had planned to. We planned our trip so carefully, but you cannot always plan for the unexpected. Go into it with an open mind. On the way home, we always leave Orlando at 8am and drive until about 8:30 or 9pm. Fredericksburg VA is a good stopping point (not sure if you go the same way, though- coming from Wisconsin). We leave Fredericksburg around 5:30am and get through NY by 9am and are home in Boston by 1:30. Both years the drive home has been great with this time frame.
In terms of drive time, it really depends. We have had bad luck both years driving South through the Northeast corridor from a traffic standpoint, but I think if we didn't hit that glitch in NJ with the van and there wasn't a major eclipse happening, the 2:30am departure time would have been perfect. Last year we picked up our rental car at noon and left at 2pm. We should have just waited and left 12 hours later because we also hit traffic in NY/NJ then. We knew we would but we got too impatient to leave. Definitely a mistake. 19 hours of driving is going to be more like 22-24 hours with stops to eat and use the bathroom. And there will always be unexpected stops. Someone will need to use the bathroom at an inconvenient time (maybe not as many for you since you have fewer people in your car), or if there is a major accident on the road and you don't want to sit in traffic, you may plan to stop and eat earlier than you planned to avoid that. Google maps will be your best friend! Expect accidents on I-95 in the Northeast corridor. Sometimes it's just someone pulled over with their hazards on and a flat tire, but it will cause back ups if there are a lot of people on the road and Google map lady will let you know!
We have 5 kids, ages 1-8 currently, so we really took a chance driving all of them down there both years. Can't live without items besides food and drinks were DVD players and lots of movies in the car, a notebook and crayons for each kid, a small white board and expo marker, some small toys for our toddler and preschooler, travel pillows and regular pillows so they could get comfy, and most importantly-small puke buckets/dramamine for kids. Also, we let them have tablets/kindles on the drive home only. That's about it. We took lots of books & extra coloring books the first year and found they just took up a lot of space in the car.
If you leave in the middle of the night, make sure you give your kids a piece of bread or something to put in their bellies. I learned that the hard way. Our 5 year old woke up from a sound sleep vomiting from car sickness because her stomach was empty and she was probably a little dehydrated. She does not typically get car sick, either. A piece of toast and drink of water after using the potty and before hitting the road is my advice. Even if they don't want it. Dramamine is nice to have on hand, but you may not need it if your kids do well in the car.
Please feel free to message me if you have any other questions. We are going again next summer, but if we can get Southwest's super cheap wanna get away fare, we may fly this one time. We will be driving again in the future though! It's the only way our family of 7 can enjoy Disney more than once in a lifetime

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