So, I'm playing the Dawnguard side. I also happen to be a Werewolf, in case this affects that.

I didn't want to become a vampire so I got partially soul trapped to get into the Soul Cairn. She says I may be able to get it back later/I'll be weak.

In there, I have the weakened soul effect. Her mom makes it sound like I turned into a vamp though. In there, I don't find anything that is part of my soul. I kind of forget to look.

I finish beyond death, leave Soul Cairn. And go 'oh flip'. I forgot to get part of my soul back. Now, I check to see if I can go back in. And I can, but I no longer have a weakened soul in Soul Cairn.

So, I don't know. Do I still need to find the part that was partially soul trapped or did I mess something up?

meh who cares, same thing happened to me

I kind of care. I want to know if I missed something or didn't explore enough.

Go west of the boneyard (where valerica is) and you'll find a building that's a bit of a maze with one of those floaty soul gems on the roof, you have to get onto the roof from the teleporter in the building, then loot the chest under the soul gem.

You didn't miss any story. You just open the chest and the you get quest complet.