First off, let me say that, like all PP, I've been there, done that, and I too, do not believe this is the right relationship for you. However, that said, here's an idea that I don't think has been presented yet.
Back off, start over. One of you move out to another place. Let her go off her BC, go to a female horomone doc and get a full hormone workup. Then you two start dating all over. Not as the "been together nearly every day of the last 7 years" kind of dates, but intentionally set it all up as if you were just getting to know each other. Go out on dates - don't go "in" to one another's homes. Once a week, not all the time. Don't text 24/7, just keep it casual, as if you were just getting to know each other.
She wants the security of an engagement before she will pursue sex - fine, treat her as a good, virgin girl with utmost hands-off respect for now. Just casually date and live separately. You try to look at her from a new perspective - not who she has been to you since you were teenagers, not who she has been through this difficult last few weeks, but as a new person - get to know who she is right now, today, tomorrow, next week. TAKE YOUR TIME. Make it clear that sex, engagement, everything is off the table, but you want an opportunity for each of you to genuinely start over and see where it takes you.
If spending time on a few dates and listening to her talk, brings out the tiger in you (if you know what I mean), then flirt with her - but don't touch. See if you courting her all over again brings out the tiger in her. If it does NOT, in either or both of you, then there's nowhere to go and no pretending. If it's EH, that's not good enough either.
If you move out, and stop playing house, and get clear that it's not just going to drag on the way it has been, it will probably open up the field of vision for each of you. What is life like on my own, is it so bad, is it pretty good, oh, look, there are other guys/girls out there who flirt with me or catch my eye! Who knew? ALso, you both need to be clear with your friends and family on the fact that you've moved out and taken a step back, and you need to be going out with your friends. You may or may not want to open up the possibility of dating other people.
Once you remove the convenience and comfortableness of living in the same house, and the known quantity of "being a couple", and you are both clear that you are NOT getting engaged, you are back to dating or courting, it may change your perspectives. She also can fully understand and live with the reality of choosing to NOT have sex until engagement (or marriage) and that's perfectly ok (I am one who is a fan of that, myself), but it's a different track and you both need to experience life on that track - no living together, no snuggles in bed, no automatic partner/date for events, no comingled bills.
I am not the dog/pet person that many people are, but you guys need to just make the call as to whom the dogs will live with and suck it up. Dogs are not children, and if you think it's hard to decide the doggie issues, and can't face that, neither of you are ready for marriage anyways. One of you take the dogs (I recommend her, she may need comfort more than you after the moving out, and it will make it easier for you two if/when you split ways permanently), and arrange for a weekly doggie date together with the dogs at a dog park or something. You don't need to offer ongoing financial support for the dogs or anything, just let her take them.
I think this isn't the right relationship for either of you. You should have someone you're just crazy about, like can't go to sleep at night because you're so excited about this person. And that person should make you feel like a million freaking bucks, like she can't wait to see you and that includes physical desire for you. That's HUGELY important! And this girl deserves that too!! If you move out, go back to brand-new dating, and intentionally court her, and the sparks aren't flying hot and heavy inside each one of you for the other, then set her free! She is scared, and probably her girlfriends are getting engaged, and she might be feeling an internal clock, but she is scared, and you need to be the stronger one here, for her benefit, as well as yours!
I would advise to just cut it off now, clean, and be done, but if you can't face that yet, then completely step back. Have her move out, give her 3-5 weeks to make other plans and you go stay with a friend or parents during that time. She takes the dogs, you guys go on a "first date" and really treat it as such. It's been so long, you may not even know what that would look like - ask your buddies. Something casual, low key, in a group, etc. TALK with her, you probably think you know everything about her, but maybe you don't. And, if she truly just needs to feel "chosen" (as it sounds like she keeps saying), then you choosing her for a date, and to give your full attention to, maybe that will spark her libido. And even if it does, don't go there physically. Keep it hands-off. Continue dating, slowly. START OVER, and give it 2-4 months max. If after that time, you're not 100% IN LOVE and CAN'T WAIT to start your married life with this woman, and she feels totally the same way, including having the major physical hots for you, then you both need to face up and YOU need to be the stronger person and call it done. I highly recommend that you include the option of seeing other people in this dating period, even if neither of you have that intention to begin with. If you know the option is there, one or both of you may find your eyes opened in ways they haven't been over the last few years.
I think you're on the right track, you're just hesitant, you don't want to hurt your best friend, of course, and you are an honorable guy. You will get to the right place, hopefully before too much more damage is done. Be strong.