The Holiday season at DLR official ends on January 6 (Wed). The Deluxe and SoCal/SoCal Select Holiday AP holders can get back into the parks for the first time on Sun (1/3) and Mon (1/4), respectively, so there is usually a bump in crowds at that point as the AP holders return to the parks to see the holiday offerings one last time. A fair number of Southern California school districts start back upon Monday (1/4), but the biggest one (LAUSD) doesn't start back up until Jan 11. Last year, Jingle Cruise closed as soon as the Holiday Season ended to have the seasonal overlay removed. HMH (Haunted Mansion) remained open through the weekend before closing to have the holiday overlay removed (unusual, but we don't know if that's going to be the "new" normal or if it'll close as soon as the season ends this year).
If you leave on the 10th, you won't be impacted by the big closures that are slated for the 11th. As mentioned, the Rivers of America (Mark Twain, Columbia, Canoes, Tom Sawyers Island, and Fantasmic!), Big Thunder Ranch, Petting Zoo and BBQ, and the Aladdin Musical are all ending on January 10 (last day). The RoA and BT Ranch area are closing to make way for Star Wars land. Aladdin is closing to make way for a Frozen show.
Overall, I think you're likely to see fairly high crowds at the start of your trip and then they might taper off to some degree mid-week before picking up for the weekend (depend on when the attractions with overlays close).
I don't think many things will be closed for the first part of your trip (1/4 to 6) because that's still during the Holiday Season, but starting on the 7th some things might go down. DLR doesn't really have a set schedule for closures, so it's hard to say what will or will not be closed when you go. The operating calendar isn't usually loaded until about 45-days/6-weeks in advance and entertainment and refurbishments are often not added until much closer to the date.