January actually isn't a popular time for free dining in the last few years. There are websites that list when Disney has had certain promotions (and when those promotions were released) for the last few years. I'm not sure I can post links here though.
From what I've seen mid-November through early December and September seem like popular times for free dining.
As far as weather with smaller crowds, definitely end of January-early February. We went Jan 31- Feb 7 and most days the weather was gorgeous. Comfortable temps and very little rain. The crowds were very manageable with only a few rides with waits longer than 20 minutes.