9 month old dog pooping in the house

I adopted a dog from a shelter when she was 6 month. She doesn't really have a tell for when she needs to go to the bathroom, we just take her out every couple of hours and after eating. She was doing ok until we had to leave town for only a couple of days. Now she is sneaking off and pooping. Here is what I am doing:

1) She sleeps in a crate first thing in the morning I take her out, stand on the lawn with her, and wait till she pees. Say go potty, act super happy and give her a treat. I let her inside, get my kids ready for the day. I try to keep an eye on her at this point, but after a week of watching her like a hawk I just can't do it with trying to take care of 4 other kids also. So I feel like it is only partial supervision, rather than leashing her to me.

2) I feed her at 9 she eats her food, I put her in her crate. I take her out after an hour nap. I have to chain her to a part of the grass, so she is forced to be on the grass. I go in and wait to see her from the window because she tends not to poop when I am around. When she starts pooping I run out, say go potty, and give her a treat. She is only chained up for a few minutes. It is the only way I can think to get her to poop since she won't poop while I am standing right there, and if you unchain her she runs to the door, and won't budge. She it is worse when it rains of course.

3) I keep her out in her dog house, or in the crate if I feel I can't give her my partial attention. I do feel like I let her in a lot to play, and with 4 kids, and two adults, she gets a lot of love. I don't trust her yet to roam, and even with partial supervision (i.e. not on the leash tether to me, like she was for a few weeks), she still has accidents.

4) I feed her dinner at 6, take her on a long walk, come home stand on the lawn till she pees give, her a treat chain her to the fence wait till she poops give her a treat.

5) I never get mad or punish an accident. Never not once. I have caught her in the act and said "echhh" and took her outside while I cleaned up, and so she could finish.

6) I have been consistent for weeks, but like I said with 4 kids I can't watch every second, and I don't want her in the crate every minute of the day. She is supper sneaky, and poops. She has had problem with being nervous around people, and with a scar on her, the vet wondered if she had been abused.

I am at my wits end, and hoping anyone has any insight. I feel like at this point I devote more time and mental energy to my dog than my own kids, it is getting out of hand.

You could try a little scold like "yucky uh-uh" (don't use no and never her name because she'll hear that a lot in people talking and you don't want to make that negative) and don't let her see you clean it up cause then she knows that's what you'll do and she never gets any discipline. They need to be told what to do in order to be good dogs. I trained my pup within days to go outside always.😊