IDK, If you can spend more time home, i would. It would fit in with the GAL but put the focus on the kids. You said you are spending time together even if awkward. They say don't believe anything they say and only 50% of what they do and i think thats applicable here. Don't push, maybe try and ask her out and court her again if she eventually accepts. She is the one that wants out remember.
Thanks for the input @KGuy
I have asked her about why she ignores me during the time I am at work versus when I am at home we do things as a family. She says she is doing it for the kids, which i appreciate. The time we have spent alone has been out of coincidences rather than desire. I will simply text her "good morning, have a good day" and she won't respond at all. I have asked her to dinner even, but she rejected the offer stating she doesn't have the desire to spend one on one time with me.