I think the reason "My Woman" has an effect is the generality and the idea of ownership.
My Car. My House. My Dog.
These are things that I own. There are millions of houses, cars, and dogs out there but this one is mine. I own it. My car goes where I tell it to go. For the most part, my house is set up the way I want it to be. My dog does what I want him to do. I am in control. So adding "Woman"--another human being with the same intelligence, wants, needs, and desires--just doesn't sound right in my ear.
Don't get me wrong--if it works for you and the woman in question, more power to you. This is something that has been debated since it became acceptable for men and women to live together without being married. My personal favorite terms was "Posselque" (POSSLQ -- Persons of Opposite Sex Sharing Living Quarters) from the 1970s census.
Years ago, I had this discussion with someone I cared about: "How do we introduce each other to the other people in our lives?"
We actually settled on "sweetheart" (e.g., "This is Maria, my sweetheart.") It denotes a romantic relationship and isn't too syrupy.