I keep getting stuck on the fact that she seems to think that the only way to show complete commitment to someone is to get engaged. And yet here you are, living with her, without sex for years even though it's important to you, going to counseling with her, telling random strangers on the internet all the wonderful things about your relationship, and even defending her to random strangers on the internet. If that doesn't scream "I chose you and only you," I don't know what does. Engagements can be broken off. Marriages can end in divorce. It's how someone treats you on a regular basis that shows you how committed they are to you, and I don't think you could possibly be doing any better.
Also, seconding what others have said... many things that seem like they will turn your life upside down really won't, or at least not for long. Humans are more resilient than we often think we are, and the house/dogs/vacations/other details can be completely changed without a catastrophe. The important thing is your long term happiness, and hers. It seems like you're both good people, and it's admirable that you're trying to work on this, but I suspect that if things don't work out, you'll both look back on this in a few years and be so glad that you decided to let go because life has gotten even better than you thought it could be.
ETA, in light of the comment below that was posted while I was posting mine: Female here. And I've never understood the push for engagement/marriage. My husband and I dated for 4 years, and lived together for 1 year, and I knew from the beginning that he wasn't big on marriage. So I was fine with being all-but-legally-married. Then, out of the blue, he proposed. Ring. Knees. Alone together at a waterfall in Iceland. The whole bit. I was absolutely shocked, and it was great because I knew that it was what HE wanted, not what I had pushed for. (And yeah, I kinda realized that I would prefer to be married rather than not, but I was so happy with him that the paperwork and the ring didn't really matter. Until he showed me they mattered to him despite his initial general resistance to the idea of marriage.) If I had pressured him to propose like your girlfriend is trying to do with you, I would have spent the rest of my life wondering whether it was what he really wanted. If she's saying she has trust issues, I just fail to see how pressuring you to propose is going to convince her that you've chosen her purely because that's what you wanted.