When's the move? 3rd / 4th week of Sept? Where I live the probability of hurricanes making landfall as a function of calendar date is remarkably low for the near entirety of the hurricane season with an pronounced spike starting around Sept 1, peaking at the 10th, then decaying over the next 10 days or so to the low-event-probability baseline.
Central Florida probably has a less-peaky, longer-duration probability curve.
Find out where your neighborhood is relative to the main transmission power distribution grid and ask neighbors what the typical restoration timeframe is. Likely <1 week. Maybe even a couple days. Use that to set your "shelter in place" timeframe.
Given your <1yo twins, if you have the space, consider:
- Buy gallon jugs of water. You'll use them regardless.
- Pack your fridge with products which can be chilled, remain unopened, and provide thermal mass to prolong the life of perishable items. Examples: 2L sodas, 1gal waters, sports drinks, beer/liquor, cranberry juice, pineapple juice, etc., ketchup, mustard, mayo, pickles, canned goods.
- Open sports drinks, pour off a couple ounces in a pitcher to drink later, cap them, then put them in the freezer well in advance of the storm. You can directly freeze half-liter water bottles as the plastic is thin enough they can handle the 4% volume expansion from liquid water to solid ice.
- Gas grill with full tank and/or extra tank. Or charcoal grill with 15lbs and fluid.
- If you need to use your range cookware on the grill, wrap the bottoms/sides in foil to keep from sooting them irreparably. Or keep a couple crappy pans you'd use for camping.
A large-volume water container is appropriate if your household water is from a well with an electric pump, because you'll lack water in the event of a power outage. You'd use that for drinking and for flushing toilets. If you're not on a well I don't know that it would be required, or even helpful, vs. filling a bathtub for flushing toilets if you have a second bathtub (which I assume you will). Most municipal water supplies and sewage systems remain fully functional during and after hurricanes, and/or have backup generators.
If a storm is imminent:
-Make sure any prescriptions for the kids are filled.
-Don't over-think it or panic, and convince northerly relatives and inlaws of the same. Millions of toddlers have lived through hurricanes just fine.
The last thing I'll add, since you're a new homeowner, is to buy a few 2x4's, a couple pounds of 16d nails or 3" T25 construction screws from a big-box store, and also get a large (25'x30') blue tarp from Harbor Freight or similar. If you have roof issues, you'll have the material on hand to do a temporary repair, regardless of whether you do it, or a neighbor, or a drive-by crew. Don't worry about having a ladder. Someone will.
Put your important papers (birth certificates, passports, homeowners policy, car titles, immunization records, social security cards) in a couple folders of different color from all the others, mark it "hurricane", and put it in a spare computer bag or backpack. Put it under a bed or in a closet, or put the folder in your desk and keep the empty bag under the bed. It will save you time if you have to evacuate.
I'm not an expert, but I've been through ~2 dozen named storms (tropical storms and hurricanes) of varying severity. I've only evacuated for one, at my wife's insistence. Have a plan, prepare, and know what you'll do as conditions change, before, during, and after the storm.