I've been there twice around that time because that's when my birthday (Jan. 20) often falls.
The first time was the Friday-Sunday of MLK weekend and the second time was Sunday-Friday (so the day before to the following Friday).
It seemed like the crowds were maybe a little lighter starting the day after the holiday, but I didn't notice exceptionally heavy crowds on the weekend, either. On the second trip, we went to MK on MLK Day and it was totally do-able. I know the general advice is not to go to MK on a holiday, but it was my birthday. I come from Illinois where the temperature is always way below freezing for my birthday and I was GOING to spend my birthday outside in 70 degree weather at MK!