Request a refrigerator in your room for medical reasons as well as a sharps container.
Make sure all medications are in labelled containers from your pharmacy. Insulin is not prescription but be sure it is clearly labelled (vials are, pens you probably need to bring in box) Syringes can be taken (mine come in bags of 10 so I just throw a couple of bags in my kit) don't forget meter, strips and lancets as well as alcohol swabs if you use them (I don't, just wash hands well before testing and injecting)
For carrying in the parks you can buy a small pack that has an freezable ice pack if you are travelling in the hot months. Cooler months I just drop it in my purse or clip to my belt in a small clip pack I got at Home Depot. (really for tools)
Always carry some fast sugar as blood sugar can really drop at the parks with all the walking. I carry a small tube of icing (no temptation to eat it unless needed) Juice at bedside in case of hypo at night. Also he is hypo and you aren't near a place for food and don't have something at hand inform a castmember who will get help
A couple of times I did go hypo and castmembers very quickly got me juice or a coke to bring my sugar up. They are great!
As to food. Ask the chef about diabetic desserts. They are available. Crystal Palace had a wonderful sugar-free brownie and no sugar added ice cream. Remember though that they will give you a bigger portion than you need. Often they will also offer fruit as a dessert and have no idea how limited the amount of fruit you can eat. Let them know you can only have a small apple or 6-7 strawberries, or a dozen grapes. The fruit platters I have been given are large enough to feed an army

If you want to go to a show where the food is inappropriate ask for something that you can eat. They are used to providing meals for special needs and very willing to work with you.
Most of all enjoy your trip