Trying to find a mod that allows you to give up kids. Yes I know I can just delete them but I'd prefer one that would let you adopt them out. For story reasons. Figured it'd been years now and someone probably figured it out by now.

I doubt,there's mod for that,because simmers were asking for It (including me) for as long as I can remember.
Anyways,this may not satisfy your wish,but here's what I'm doing to ,,put for adoption'' any unwanted child in hood:
-Create a ,,social worker'' or whole personel,lol.
-Place her on a large lot with a spacious house;
-Whenever you spot a family with over-excesive number of children-switch back to your ,,social worker'',visit that family (use MC or testingcheatsenabled true-select right mouse button-add to active family and VOILA! :D),bring kids to ,,orphanage''\remove parents from their family trees with MC (I am sure you are able to do so with It).
-If you decide that you want certain boy\girl to be adopted-do the same as above,just with the family of those willing to adopt.
It's neat when you actually can communicate with youngsters before bringing them home.