Disneyworld is one of those places where you won't be lonely if you dine alone. The staff will take care of you and give you as much extra attention as they can spare the time. MY choice would be Garden Grill in the land. It's all you can eat, but the food is served to you so you don't have to get up and dodge people to get food. The restaurant revolves, so there's something different to see and hear throughout your meal, and it has good character interaction - Farmer Mickey, Pluto, Chip & Dale.
On Christmas day, I agree with the others. If you must go to a park, then go to EPCOT. Get your priority seating for early evening, so you can get a place to watch illuminations early - even though it probably won't go off until 9:30. Stay to the very end and then stay a little longer until the crowds clear and the bus lines have had time to go ddown a little.
You don't say where you are staying, but if you have the choice and budget for it, I would stay at an EPCOT resort or Swan/Dolphin.
There are lots of things to do during that time of year besides visit the parks. The resorts are all theme trimmed in the main buildings. Sometimes there are evening performances at the resorts - I saw a handbell group at Port Orleans Riverside once, for example. Then you have the regular "shows" at resorts like YeHaw Bob in the lounge at Port Orleans Riverside.
You can tour the resorts - and I would definitely make time for that. Some of the trees and gingerbread displays are nothing short of amazing. Grand Floridian, of course, Animal Kingdom Lodge (Jambo House), Wilderness Lodge, Yacht Club and Beach Club top the list. Coronado Springs. Some of those resorts have tours, and they're free. I guess I'd make my initial resort exploration early in the trip and stop at each lobby concierge for their "calendar of events" so you can plan the rest of your trip.
Offered only at Christmas time is a special pay tour about holidays at the world. If you want to take that, BOOK EARLY! because it sells out.
Oh, and Winterland/Summerland Miniature golf is a hoot. Thirteen days is a long trip - unless you have a parkhopper as part of a package, you might want to get an Annual Pass. They're pricy but thirteen days is past the break even point and the AP gets you 10% discount on some shopping and food. A list of perks will come with the pass, but you should always inquire.
Good luck! Merry Christmas!