Hmm I think I will be the contrary voice ...but I would not rush it. I definitely hear your frustration, especially being 8 mo preggo with twins! I remember how overwhelmed/tired/bulky/billion thing list-ed I was 8 mo into a normal singlet preg.
From reading your post, it sounds like your MIL loved you FIL very much. And I can only imagine how many hours and hours and hours she spent preparing all those boxes. I imagine a grieving woman over an ironing board. With each sleeve, maybe she was remembering her DH in this outfit, or what he said to her in this shirt, or something that happened in this suit. It sounds like she didn't much warn you of the truckload of memoiries? Maybe because she would not/could not take a refusal, even a gentle one.
How about a consignment store? Make an appt and bring what you can. Or is there a young teenager on your block in need of extra cash, someone you could hire to help out? Maybe though, taking your time would be okay?
It would seem to me that your gentle care of your FIL's things will be the greatest gift that you could give your MIL at that time. And you are a hero for that! In whatever course, do be sure to take care of your twins and yourself...lots of fluid, lots of rest. Even if it doesn't get garage-saled soon, the truckload took a life-time to accumulate, surely it will take awhile to sell/consign/share.
BTW, some cities have organizations that will take business clothes and "repatriate" them to persons needing them to get out of unemployment, first job interview out of trouble etc.