Don't know if I can help, but it's my understanding that Brown Turkey has a small eye which has a reddish color from a very early stage, and yours don't exhibit that.
Yours make me think of my Kadota based on the appearance of the figs, but I'm not certain about the leaves.
Someone else will probably chime in with better help.

I thought Kadota orginally but my kadotas are a bit more yellow. A turkey tends to turn brown-purple towards the end of ripening.

What part of the country are you in?? The hot desert? The humid south? If you are in the desert, maybe you
arenot giving enough water. Water dailey and see what happens. You can't always count on a nursery plant being what they say it is. Maybe your tree is diseased. Have you sprayed
any weed killer in the area? You can spray 20 feet away and wind will carry enough herbicide to kill off leaves. Lots of possibilities. More info needed. FF

I am in Houston, Zone 9 as stated above. We have had plenty of water and there has been no weed killer sprayed.

The leaves fall off because of rust. The air is just too humid in Houston. With all the rain, the problem is even more so this year.

There is milky sap on the stem end when you pick the fruit because the fruit is not fully ripe yet.

If you bought the trees at a nursery in Houston, there is not telling what you have. I say this with lots of experience. The more popular varieties around the Houston area are Celeste, Kadota, Alma, and Brown Turkey. You already said they do not turn brown, so we can eliminate BT. If your fruit stays yellow, then I would tend to agree with everyone else and say Kadota (or one of its variants). If your tree has tinges of golden brown, it is probably Alma. The fruit on your tree does not appear to be the same size/shape as the Celeste variant that seems to be everywhere in Houston.

If you live close to the Clear Lake area, I can direct you to where a Celeste, an Alma, and a Brown Turkey live and hang over a fence in public view. You could go to those locations to see what they look like and what stage of development they are in.

~james

I live in Brookwood so I am in Clear Lake. I bought them at Mas Nursery in Seabrook. They said they were BT. Well I haven't seen anything that looks like the rust I have seen as examples on my tree but maybe I don't know????

Rust is apparent on several leaves in your fourth photo. Go and look closely.

Sorry, I think the third photo.

Ok with that said and you see rust, what do I do to get rid of it? Or at least help remove it...

You can read the linked post in this forum; chances are a search of the forum will yield more info.

Here is a link that might be useful: Discussion of rust...

Maas Nursery claimed they've only had two varieties of figs go through their nursery... BT and Celeste. I asked a few years ago when I saw a 1 gallon tree there that didn't look like either. I bought it thinking it might be the famed "Banana" variety fig that is thought to have originated in Seabrook. If it is, I am in agreement with everyone else who has said Banana is very similar (if not identical) to Kadota. Another give away for Kadota over the other varieties is that the fruit is in tighter clusters. It is something I see more in my Kadota trees than the other trees.

~james

If I had to guess, these look like gold figs. Gold figs ripen in a bright green color, from the photos, the leaves and the figs are a bright green. I'm just learning about this stuff myself, so I might not be right.

You are picking them unripe.
Let them ripe till the skin shrink a little and the fruit is hanging down,in vertical position.
You got a Kadota there,and the climate is adeqvate to ripe Kadota in Houston.