Does the Florida government have programs to help people in this situation? It would be much cheaper thant the cost of trying to rescue them later.
When you go to shelters you are told to bring your own food, clothing, diapers, water, toiletries, flashlights, blankets, pillows and quiet things to entertain your children without electricity, because shelters can and do lose power sometimes. If you don't own some kind of luggage that's really difficult to manage on a bus with several children, even if there is a bus. Lots of people figure that their kids are better off in their own homes where they will not be as frightened, though depending on where that home is in terms of a water body that could be a deadly mistake.
Most authorities in Florida right now are telling people that if they are on high ground in a building built to the new hurricane code, and do
not have a medical condition that means they cannot survive without electricity, then they should stock up on storm supplies and at *least* a full week's supply of shelf-stable food and water for every person in the house, but stay off the roads. However, if that doesn't apply to your situation, bug out ASAP or by early Tuesday evening so as to be in a high-ground shelter or out of the path by the middle of Wednesday.
My childhood experiences of hurricanes took place in the old days when evacuation really was not a thing, and we simply didn't leave. However, we turned our elevated homes into fortresses, with plywood on every window and door, steel straps bolted into all the roof trusses to anchor them to the walls, and the inside of the house rigged up as a kind of campground with stocks of canned food, kerosene lanterns, bottled water and powdered milk, and tons of tools, tarps, ropes, wood and sandbags, in case something breaks and has to be repaired in the middle of the storm. You lay the kids down to sleep on a pile of blankets in a bathtub with a mattress over their heads, the adults stay up all night keeping watch, and you have an inflatable dinghy at the ready just in case. That kind of preparedness is less often done nowadays, when people in storm zones who have the money primarily clean out the fridge and crank down the shutters before hitting the road.
The standard declaration given by first responders to people who choose to ignore "mandatory" evacuations is good luck and God help you, because we won't be risking our lives to do it. You're on your own from the point that the road into your zone closes until the storm has fully passed.
If you are willing to stay at a public shelter (and some people are not), most of the cities in Pinellas, Hillsborough, Sarasota and Manatee counties are running shuttle buses to take people there today and probably for at least the morning tomorrow.