From what I understand about trees and dams, there are two things that happen that make it a very bad idea to allow them to exist on the dam. One, the roots dig through the dam to the water. When the tree dies, the roots shrink and then rot, allowing water to follow that path. Second, is that animals like to tunnel along the roots of trees to make their dens. Beavers are the worse, but rats and nutria area also very bad about this. Probably some others too.
Trees look pretty, but they slow down the amount of runoff you get. Compared to an open pasture, I've figured it's about a 3 to 1 ratio of how much water the trees catch. That's just my guess from what I've seen at my place.
I don't think the trees are drinking very much of the water. Nothing like what you are losing. I think you have a leak.
Bad soil might be a big part of it, or it might be roots through the dam. The first thing you need to do is let it leak down. Once it stops leaking, you will know at what level the leak is. Doing a very detailed search might allow you to find it. If not, then betonite might work. Read up on how to install it. There are some different versions of how it needs to be worked into the soil, mixed and compacted.
Not knowing where you are in East Texas, I can tell you that in and around the Tyler area, sugar sand and iron ore are mixed into the ground all over the place. You might not have sand, or know of it, but if you go down a little ways, you might hit it. I found a pocket when digging my pond. Sand is a very bad leak, so I dug down several feet and filled with good clay, then compacted it. I got lucky and it worked. If I hadn't of hit that sand and it was just a few inches under where I was digging, it would have been a leak that I might never of figured out. Same thing if there is iron ore just under the surface. Nothing you can do to stop it from leaking other then packing clay over it or digging it out enough to fill with clay and compacting it.
Liners are not practical and nothing else will work. You will need heavy equipment to fix this. The question is how bad is it and how bad do you want it fixed?
Eddie