I think the biggest benefit of going right after New Year is that the parks are still decorated for the holidays. The AP holiday blockout ends around January 4, 2020. Now that could mean the holiday season is going to end on January 5 (in DL) and potentially January 6th in DCA (assuming they do the Three Kings Day celebration again in 2020). If you went right after New Year (before the weekend ends), you'll probably get to see the decorations, ride overlays, shows, etc.
In terms of March, are you asking about 2019 or 2020? If you're thinking of going this year, you'll probably miss the largest concentration of the Spring Break season crowds; next year, maybe not so much. F&W should be happening; it's slated for 3/1 through 4/23 this year. While it has been happening annual for several years now, it's hard to say what will happen or not happen from year-to-year (especially with the opening of SW:GE slated for this summer). Likewise, the Easter Egg hunt has been happening closer to Easter the last few years (two weeks or so before Easter), so you might miss that if you go in late-March this year. Also, we don't know when F! or WoC will be returning this year (it just says "spring").
There will always be crowds at DLR, so I'd be hesitant to try and guess which would be busier. So, I guess it comes down to the Holidays versus F&W and which you think you and your family would enjoy more.