Hi there, Bella, and thanks so much for checking in!
Two times of the year will be quite suitable for you, I think, in terms of weather: from after Thanksgiving through mid-December, and (my personal favorite time to go) from January 3rd through just before President's Day weekend,
The weather in Florida at those times of the year is glorious. In November/December, you'll have clear skies, daytime temperatures in the 70s/80s, with nighttime temperatures in the 50s/60s. In January-February, the weather is a bit cooler, but can range up into the low 80s. Nights are perfectly cool.
Of the two, the value season would definitely be January-February. Resort hotels are offering good deals at this time of the year, and restaurant availability tends to be reasonably open You'll also find crowd levels to be considerably lower then than in November-December (lots of people come for the Christmas decorations and atmosphere).
As for accommodations? Each of the three classes - Value. Moderate, Deluxe - offers a king bed option. Check out
Disney's Pop Century Resort, a Value Resort that explores decades of popular culture, and
Disney's Port Orleans Riverside, an option that would actually provide a trundle bed in addition to the two double beds (you might lose this feature in a King room).
Please feel free to let me know any more details of your trip and I'll be glad to help you narrow down your choices further and create a wonderful itinerary for your traveling party.
Again, thank you for writing!
See you later, celebrator,
-Deborah B