I left my husband when my son was just a baby.
Going back and forth can be really tough, but things do get better.
I think your daughter's behavior might be completely normal. Switching gears from one environment to another can be hard. It sounds like she needs some time to decompress and it comes out in tears. She's letting it out.
I'm not saying you do this, but make sure not to barrage her with questions as soon as she gets home. Did you have a good time? What did you do? What did you have for dinner? They are perfectly innocent questions, but little kids can feel like they're being interrogated. Give your daughter time to just chill out or cry and let her talk about what she did when she's ready.
She could just be tired. I know that my ex, for some reason, felt that they had to go-go-go the entire time. My son came home worn out. He'd have a little melt down and want to lay down on my bed with my pillow. I'd give him some time to settle down and then his older sister and I would do something silly to make him laugh and all would be right with the world again.
Instead of seeing it as "damage control", perhaps try a different tactic than you are using. Give her a hug, tell her how happy you are to see her and then give her a little bit of space. I've dealt with a lot of child therapists and they say, especially for a child this age, to avoid asking "why" questions, because very often, little kids don't know "why" they do something. If you ask her why she's crying, she may not even know.
My son just turned 17 and I can tell you that things do get easier. The older my son got, the more he had some say in things. If we had family in town and he wanted to see them instead of going with his dad, his dad was fine with it. If he wanted to go fishing with his dad during salmon season on my weekend, I was fine with it. Especially now, at his age, it would be ridiculous for his dad and I to argue over such things.
Little kids feel like they don't have any control over their world. They don't have great communication skills when it comes to adult things that affect them. They all deal with emotions differently.
Let Gabby know how loved she is. Work with your ex to make transitions as easy as possible, and give her some time in a buffer zone to get reacclimated to things when she comes home.
If you are deeply concerned, you can contact a therapist that specializes in children of divorce. They can offer a wealth of support, not only to the kids, but to the parents as well. When my son was little, he had an amazing counselor at his school and she helped him so much. She helped me a lot too because I was dealing with an ex who didn't believe in counseling which is why marriage counseling never worked for us.
Hang in there. I wish you the very best.