How about My Husband's Penis Won't Fit? The FL is not conventionally weird, but she can come across as a bit odd from time to time, esp in the first half of the drama, imo. I didn't find her weak per se, but she does flow through life with an air of equanimity that I found rather interesting ...
Boku no Yabai Tsuma The FL is weird and crazy all at the same time. The ML isn't far behind. This drama was a really fun watch, esp if you like the mystery, thriller, suspense genres & dramas with a lot of twists and turns!
Both of the leads are pretty odd already but then they add in the body swap. It's pretty hilarious and really underrated imo. You just have to remember that the show isn't taking itself seriously so, as the audience we shouldn't either. I personally loved it.
Date (Japan) - she is an economics expert, socially uninterested, wants a husband to have children. He is an "educated idler" and wants someone to continue to support him after his mother passes away. Both are a bit odd looking. Reviews indicate that this becomes a very romantic drama. I've only watched the very first, saved for another time.