I(30-NB) love my partner(31-NB) but Idk if love is enough anymore

I honestly don't know what to do anymore, so I'm hoping for some outside perspective.
My partner(31-NB) and I(30-NB) have been together for 6 years and have lived together for 5. We have two cats together(1 who is very attached to them, the other equally attached to both of us), rent a house together, and have even been talking about buying a home. I really do love them, they’re my best friend in a lot of ways. I feel safe with them, respected, and like I can completely be myself. They're kind, thoughtful, and they show love by taking care of me and doing things for me. We never raise our voices at each other and we are very sweet with each other, do all the domestic things together, no problems with cooking or taking care of the house and we spend lots of time together.

The problem is... I feel completely starved for physical affection.

A few years ago they had a hysterectomy that had some really awful complications. Ever since then, they've had almost no interest in sex or physical intimacy. They never initiate. If I don't, we can go forever without anything happening. We haven’t been intimate for almost 9 months.

They'll cuddle me or hold my hand if I ask, but I can tell they're doing it because they know I need it, not because they actually want to. But I don't just miss sex, I understand that it is painful for them, and they have little to no libido, and I would never ever pressure them into it without their full and enthusiastic consent. I just want to be wanted.

I want someone to reach for me because they want to. I want someone who's excited to kiss me, cuddle me, or just be close to me. Instead I feel like often affection happens because I asked for it.

We've talked about this so many times over the years. Year before last we actually broke up because nothing seemed to change. I waited for months hoping things would get better, but they didn't. At one point I even suggested that if they truly had no interest in sex anymore, maybe I could seek that elsewhere while we kept our domestic life together. They weren't comfortable with that. So we broke up, I moved out and got my own apartment by myself. But before I moved out, after the security deposit was settled I asked if we could continue our relationship together, even while living separately. I was terrified to lose them and I felt that with some time and distance we could both work on what we needed to work on. They were amenable to this and we stayed together while I lived separately.

I moved back into the house after a year because being apart was one of the worst experiences of my life. I was miserable, broke, and I really believed maybe we'd both grown enough for things to be different. But now it feels like we're right back where we started.

They always tell me they're trying, and I believe they mean it. I don't think they're lying to me. But I also can't ignore that nothing really feels different. I worry that maybe they aren’t trying to change, but to avoid losing me. Another issue is communication. I'm always the one bringing up hard conversations. They'd rather avoid them and hope everything smooths itself over, while I feel like I'm constantly carrying the emotional weight of talking about what's wrong.

The thing that makes this so hard is that I don't think either of us is a bad person. I think we just might not be compatible anymore.
I want a relationship where we keep choosing each other. Where we still want each other years later. I don't need constant sex, but I do need to feel desired. I need someone to want to hold my hand, cuddle with me, or pull me in for a kiss because they actually want to, not because they know it'll make me happy.

I find myself pulling back on my affections. I don’t reach for them as much, don’t seek them out for kisses as much, try not to act too excited when I see them cus I thought maybe that would make them more comfortable? Like there was less pressure on them? Like I’m trying to learn how to be less physically affectionate, and it makes me so sad.

The thought of leaving scares the shit out of me. Financially I don't know how I'd manage. I don't know what would happen with our cats. I still love them so much.

At the same time, when I imagine leaving, I also imagine finally getting to be independent, chasing goals I've been keeping to myself, and maybe one day being with someone who genuinely wants me the way I want them.

I'm ashamed to admit that I've gotten so lonely that I've caught myself thinking about cheating just to feel wanted again. I haven't done it, and I don't want to. I could never betray them like that. But the fact that I've even had those thoughts is what made me realize how emotionally and physically empty I've been feeling.

We have discussed couples counseling, and I thought we were both on board. We both started going to individual therapy (as a prerequisite to couples therapy) and I have been going regularly for 1 1/2 years, I even do a weekly DBT skills group. But once the copays got more expensive for them after only a few sessions they just… stopped going. And the conversation is shut down whenever I try to bring it up because it’s suddenly too expensive for them…

So... am I expecting too much? Is this something people can actually work through? Or am I holding onto a relationship because we love each other, even though we can't give each other what we need anymore?

And if my partner sees this... I'm sorry. I'm not trying to embarrass you or make strangers take my side. I love you, and I know you've been through so much. I'm just feeling lost, and I need perspectives from people who don't know either of us because I can't tell anymore if I'm holding on out of love or out of fear.

TL;DR: I've been with my partner for 6 years, and while I know they love me, I don't feel wanted. They rarely initiate affection or intimacy, and any physical touch only happens if I ask for it. I don't just miss sex, I miss feeling desired and chosen. We've broken up over this before, got back together hoping things would change, but we're back in the same place. Am I asking for too much, or are we just incompatible even though we love each other?