Hello, OP.
I cheated on the father of my firstborn, who I know now is asexual (we weren't technically together when she was conceived but he had been drinking) because he chose to stand by me but it wasn't what he had ever wanted and unless I practically forced him (wince) or he had had a drink, there was no intimacy. I felt unloved, unwanted, ugly, you name it. I would sleep with the odd person now and then because I needed to feel wanted and loved. The reality was, that it was him I wanted to be intimate with and it didn't make me particularly happy.
My mum went through early menopause and has slept in a different room to my dad for a good 15 years, might be closer to 20, I'm not sure. I know it affected him badly because he went off the rails a bit and he also shared some TMI (eww, I didn't want to know my mum wouldn't 'put out).
And my used-to-be-rampant libido has chilled out over the last six years, seems to have been since they changed some of my meds but I also had a relationship with a sexually coercive arsehole who objectified me constantly plus I carried a hefty mental load. I've discovered since then that there is a direct link between the mental load I carry, plus the household chores and childrearing, to my libido.
Typically, I'm now with someone amazing but he is very low in libido and is also very self conscious, he has put back on four of the stone he previously had lost eight of, since we have been together.
I do feel unwanted, ugly, fat, etc. But I am also more understanding having been on the other side of it.
I would write her a letter. You tell her how much you love her and what for. And what you love about what she does and for bringing your DC into this world and feeding them. You explain your need for intimacy. You ask her what YOU can do to relieve her of any burdens which may be affecting her libido.
She sounds 'touched out'. Kids are usually very physical and needy and noisy. My nearly eight year old is all those things and leaves both me and her stepfather feeling flustered and touched out.
And definitely mention couples counselling. If she is unwilling to try the counselling and action what I mentioned above, then say you have to walk away so you don't hurt her by seeking affection elsewhere
For me, it wasn't about the sex, it was the intimacy, contact and connection with the other person. And you do expect those things from a life partner, with or without penetrative sex.