You might want to post this on the Dsabilities board; there are frequently threads there about taking insulin, syringes, your glucometer, etc. on planes and into the parks.
I'm an insulin dependent diabetic too. IMO, you should just take your vials of insulin and syringes with you, and not pre-draw. You can get a cryopack thing to put in a small bag or backpack to keep your insulin cold. The first aid stations at the park also will refrigerate your insulin for you, but it can be a hassle to go back to the station to draw it up if you're on the other side of the park.
Since you're newly diagnosed, you should plan on testing your blood sugar a few more times a day while you're at Disney; all of that walking can make your levels go lower (conversely, eating at the parks, away from your regular routine/schedule can make it go higher). You may need to adjust your insulin amounts according to what your blood sugar reading is.
There's no problem at all when they check you go through the bag check area at WDW. They're looking for weapons and contraband; they see medications all the time, and they'll wave you right through. If you're flying, pack your insulin and supplies in your carry on; security there will not make an issue out of your syringes as long as you have the insulin vials too.