We went one year over MLK weekend in Jan. Only MLK day itself (Mon) was bad, we were at AK and it was like the first year AK was open which may have made it seem more crowded.
We went one year the first week of Feb. in value season.

It was heaven, low crowds and low resort rates! But took our kids out of school for 5 days. They were both in elementary so it wasn't too hard on them.
Went last time the last week of Feb. which was the week after Pres. Week, it was Mardi Gras week. Also a great time crowd -wise, but paid "peak season" rates. Don't remember the crowds being much of an issue, especially if you use fast pass, emh or other crowd management strategies.
Before having kids, DH and I did a day trip to MK from Melborne, Fl. during Easter week. We had no clue, did everything wrong and suffered the consequences!
This year we are going back again the last week of Feb. because it is school break, luckily it is the week after Pres. week again. My kids are missing one day of school so we could decrease airfare by $100 each ticket. One day is about all I would let them miss now as they are middle school and there is a lot of work to make up each day!
Good luck!
