I know exactly how you feel. Especially about the children. None of us signed up to have children only half of the time. I totally, totally get it.
But if I may... pushing these R talks simply IS NOT HELPING. Maybe she finds an apartment. Maybe she doesn't. It is all completely out of your control-- unless you apply more pressure, more R talks, more more more... that may be solidifying her decision. The more she hears you love her the more she feels guilty that she doesn't feel the same way back.
What can you do to STOP talking to her about this stuff? I get it that it is triggering to see her looking at apartments. it doesn't mean she's getting one. I think this is one of the reasons that GAL is such a big focus here-- you need something to take your mind off of your W and put it back on you and your kids, and honestly to get physically away from her so you aren't peering over her shoulder at her computer and freaking out about what she's looking at.
So, can you make a plan for the next few days? Things for you, things for you to do with your kids? Get yourself out of the house so you can avoid putting yourself in a situation where you feel compelled to say something? And, have some strategies for how to deal when you are triggered and really really really feel the need to say something to her. When I felt like I couldn't stop myself from saying something to my H and opening up an R talk, there were a bunch of random things I would do to stop myself, ranging from opening and closing my hands over and over and repeating a mantra in my mind (ranging from "he's an a-hole that I don't want to live with anyway" to "amor fati") to hugging my kids or coming to this site or calling a friend. Eventually, it got easier. (And I need to start this back up again.)
And not that my sitch is anything anyone would want to emulate and we are not in piecing, but FWIW my H looked at apartments and found the perfect one and put in an application and got it. And that was what turned him back around (for the time being). But he had to do that 100% on his own.
I just think the more time you can spend focusing on yourself and what you need to calm down and re-center yourself, and the less time you worry about whether or not she's really asexual or is going to finally follow through on her threats to MO, the better. One thing that helped me here was focusing on my kids. It sounds like your W is going through a very rough time, no matter what, with the drinking and the rest, and your children are going to need you at your best. So do what you have to do to pull it together for them.
You can do this, H.
Me (46) H (42) M:14 T:18, D9 & D11 4/19 - 12/19: series of escalating BDs 9/20 - present: R and piecing