Welcome to the forum
it sounds to me like you might benefit from some counselling to perhaps; catholic school, health issues, cheating. All those alone might be difficult for some and with poly on top, maybe even more. Couldn't hurt anyway.
First off, you've only known this woman for three weeks? That isn't near enough to trust someone who is a new partner to yours. That could take up to a year or more. You don't have to love her, you should probably not even try to. Certainly don't expect that you will be sexual with her. Its not a requirement. So she loves you. That's nice, what an added bonus. Thank her and do your thing. Its great you can see you will even be good friends. Just be yourself.
I don't see why you would necessarily feel loving in this situation. Its new and you have a history of cheating with this guy. Give it a chance, take your time. See how you feel at three months or more. If nothing changes and your trust still isn't there enough or you just don't feel love then I would suggest you have just settled for them or will never trrust with them. No biggy, move on. You have a whole life ahead of you and lots of other things to do.
Being on top of your love and being what you "think" you should be is not necessary. Just be. Let whatever it is come. Leave it all be and relax into it. When you do this I would think that you will see how you feel and how your body reacts to the situation.