Girlfriend 30 F broke up with me 28 M 2 weeks ago but we still love each other and live together, what should I do?

My girlfriend (30F) and I (28M) broke up two weeks ago. She said she'd fallen out of romantic attraction toward me, partly because when I drink too much I can become silly and sometimes blackout. We both drink to this stage but me little bit more often and we’ve both said we need to learn to control ourselves.

Over the past year I've also been drinking more to deal with anxiety, and I've struggled more since my dog died about 10 months ago. I know I need to change my relationship with alcohol and work on myself, regardless of what happens with us.

The confusing part is that we still live together and genuinely get along. We haven’t even had an argument. We laugh, help each other around the house, text/talk about our workdays, more now then we did when we were together. and still say we love each other. She initially slept in the spare room but recently has chosen to sleep beside me again, and we've had some cuddling.
She also kissed another guy while drunk shortly before breaking up with me. I'm not angry at her and I still trust her.
I don't want to manipulate her or pressure her into getting back together. I want to give her space while showing through my actions that I can become a healthier, happier version of myself. I'd love the possibility of rebuilding our relationship someday, but I don't want to chase her or play games.

What would you do in my situation? Is staying close like this healthy, or should we create more distance? And how do I give her space to miss me without deliberately withdrawing?

TL;DR: Girlfriend broke up with me 2 weeks ago after losing romantic attraction, partly due to my drinking. We still live together, love each other, laugh, cuddle sometimes and get along well. I want to work on myself and hopefully rebuild things someday without pressuring or manipulating her. Looking for advice from people who've experienced something similar