The timeshare you are staying in probably has a ticket counter, and you might get a good deal there. They will also probably have periodic buses to the various Disney Parks, but though you will save your daily parking fee, the shuttles take quite a while to get there (multiple stops usually) and only come and go at wide intervals. You will have to decide whether that is worth it to you.
Bring water cups/kleen kanteens/whatever and get ice water everywhere. Drink nothing else. It's going to be hot and sweaty, and no drink besides water will satisfy you enough anyway.
Bring a smaller soft-sided cooler or two in your stroller (or instead of a purse), and have the adults bring backpacks, either with small coolers inside or with non-perishable snacks and fruit.
I have been to Disneyworld several times and have never had any problems leaving coolers, bags, etc., in a stroller outside of rides. I would never leave a purse, wallet or camera, but once accidentally left my camera (visibly) and it was still there when I returned. Whewsh! For rides where you can't take a bag, there are always rental lockers right there (fairly cheap), so that you can lock up precious items. That is rarely required, though.
Do bring ponchos, because chances are, it might rain and you will be happier to bring them than to find yourself in great need when they are $8-10 each. As a tip, dollar stores sometimes sell 2 ponchos for $1 - definitely worthwhile looking into! Occasionally, it rains cats and dogs, like extremely heavy, for an hour in Orlando, and then the skies clear and it's a beautiful evening the rest of the night, but if you are soaked from that downpour, then it will be harder to enjoy the beautiful evening. Also, you can use the ponchos on wet rides and not have to worry about your purse as much.
Since your lodging is going to be paid for, I'd say you are extremely fortunate! That will really help.
Often, when we go for long trips to Disney, we leave one day free and instead of paying for park passes, we spend the day at Downtown Disney, which is like a shopping center with all sorts of free activities for the kids. There is the Lego Store, which has a huge area outside to play with Lego, multiple tables and dozens of people playing with Lego. There's a little slide/clubhouse made of lego, too. One of the Disney stores has areas to play with Mr Potato Head toys and to play with these little Mr POtato Head video games. There is a small train ($1-2 per ride) and a carousel, so that little ones can get rides in, too. There is a water splash/fountain area the kids can play in. When mine were little, they totally considered Downtown Disney a theme park and destination. Plus, I bet your timeshare location will have tons of activities at it, and multiple pools, too, all for free.
Paying extra for "park hopper" tickets seems useless to me. To go from one park to another the same day is just a waste of valuable ride time. Take one day for each park (or maybe two for Magic Kingdom, that place is fantastic) and no more. Epcot is probably not a great park for such little ones, so you might want to skip it this trip.
Be sure to check on Disney.com to see what the hours of the various parks are, because they can vary drastically. Some days, Animal Kingdom closes at 5pm this time of year, while Magic Kingdom is open until 10pm. There are fireworks and a parade at the end of the night at Magic Kingdom, which is a fantastic spectacle, but you know your children best and whether or not you think they can make it that late, and whether or not fireworks scare them.
Oh, even though we often bring our food and snacks, we usually buy a refillable popcorn bucket at the park (or bring the ones that we bought years ago) and refill for cheap ($1? $2?) whenever the kids want a special treat. It makes my children feel like they are getting Disney-food, instead of just pb&j sandwiches.

Have fun!