First welcome to the forums. This is a great place to learn.
As far as your problem. For sure it is the rods and linkage that control the handle action. Look at this link and it show a pix of 64/12 - 66 but I think youll get a better idea of what is inside the door.
There are rods that pull on the latch when you lift the handle. Yours are almost certainly off.
The link comes from Mustangs Unlimited which is a great place for parts.
I am not saying you need parts yet, but they are very good Mustangs Unlimited - '65-73 Mustang Parts & Accessories Online Virtual Catalog: Page 188
Hope this helps a little...... Print Dad