Honor my heart, or satisfy my urges?

I'm struggling with a very difficult dilemna. I've met a woman who is smart, independent, very deep, and sensitive, like me. She is also very attractive. To my great surprise, she find me attractive, too, and we have gone out a few times. Recently we had sex. However, I do not feel emotional connection, emotional safety, with this woman. I feel less and less comfortable sharing deep things with her. It's not that she's "bad," or "malicious," in ANY way; not at ALL. She's just not the kind of gentle woman I'm looking for.

A bit of background: I waited for many years for marriage to have sex. This felt 1000% right to me— emotionally, not for religious reasons. Well, marriage didn't come. Around when I turned 30 (I'm in my late 30's now), I finally broke those chains, little by little, until they were entirely cast off, because I was so sick of living with the weight of a vestige of a dead dream. I finally did it with a woman who I haven't seen since, and for whom I didn't even feel love. It breaks my heart.

Since then, I've had several partners, including this recent woman. However, touch... let alone sex... is something that's very precious for me. I am not comfortable giving it to just anyone. Ideally, I would love the person first. But since that takes so much time to develop, and there is no one on that horizon now, I've chosen to satisfy my physical desires even when my emotional needs aren't met.

Which is why I'm so torn and hurting now.

I want to say to this woman: I don't want to be with you. I don't feel connected to myself when I'm with you. You're not right for me. But I know that if I do that, I will be giving up a very rare chance for sexual interaction with an attractive woman. (I refuse for legal and ethical reasons to go to a prostitute.) There IS another girl I know who wants to be sexual with me, but she's just as dumb as a log (sorry), and I'm not even attracted to her. So... I'd likely be high and dry again.

Then again, sex isn't even that important to me right now. My emotionaly needs are SO much stronger than physical ones. SO much more real.

I know the answer to this dilemna is to tell the woman that I don't want to be in a sexual relationship with her, and just want to maintain the friendship. I believe she's open to that. But I have so much trouble letting go of my physical desire.

What will save me before this situation crashes and burns?

How can I accept giving up touch from a woman?