I can definitely relate to what you're saying, probably any pilot on here will tell you the same, there is a time when you mesh with the airplane in a sort of way that is hard to explain, you kinda know what the plane needs or wants without it asking you for it, kinda like a bit of thrust when its losing a little energy, more drag otherwise you'll be left to high or too much sinking when on an approach, the term "flying by the seat of your pants" comes to mind, which by the way was the toughest thing to understand back in my flight school days probably because I didn't have the experience nor the knowledge to just Feel at one with the Aircraft as your title says.
I have been on the Airbuses for the past few years and I have found it a little more difficult to become one with the plane though, In my opinion it's probably because of the FWB, non moving thrust levers and no feedback on the side-sticks that have made me feel that way, having flown Boeing's and MD
's that had no FWB, for me it was much easier to "feel" the airplane.
Also I think that "Felling at one with the plane" may not always be flight related, there also comes a time when you feel really comfortable with the airplanes procedures (normal and abnormal), checklists and cockpit flow scans and everything else.
Just my opinion.