The good news for your budget is that the ARRRcade on the Dream is gone.
The bad news is that there is now a candy shop/sweet treat shop called Vannellopes. This shop has a lot of different candy, treats that are exlusive to the shop, gelato and ice cream. It costs extra, so you might want to budget some for that in place of the arcade. If you have sweet lovers, not that I have that problem...oh no...except for Laffy Taffy. And Tootsie Rolls. And did I mention gelato?
I would suggest budgeting for small gifts in Nassau..they have a shopping area as you get off the ship and you have to walk thru it again to get back onboard. Surely, you will find some magnets, mugs, pens, etc there. Bring cash or a credit card for these places as they don't take your KTTWC for payment.
On
Castaway Cay, you can use your KTTWC for purchases on the island with the exception of stamps (cash) and if you do excursions that are run by independent operators thru Disney (parasailing, wave runners, etc) you might want to be bring cash for tips for those operators if you are so inclined.
Smoothies, alcohol, Vannellopes, Movie theater candy, canned sodas etc are not included in your trip fare. Some items in room service are also extra and you would want to have cash to tip for any room service items you order. We typically don't do room service if we are on deck 9 or 10 as it is a quick walk up to Deck 11 for fountain drinks and all you can eat soft serve ice cream.

And other snacks.
Just some thoughts on other items in addition to the excellent advice above
Have a great first cruise.
Oh and bring some funds for booking your SECOND
Disney cruise on board so you can get a discount and OBC for the future cruise of 4 nights or longer. You can always book a placeholder for 250.00 if you are not sure of when you might be sailing again. Then you have 2 years to decide and book .