I did your same process with the T pins and the three layers, however, I used safety pins to hold the sandwich together. I started in the middle and worked out to all edges evenly. I pinned with the smaller pins about every 2-3 inches. It took a lot of time, but it didn't shift or skoot. When I quilted it, I started in the middle and would guide with my left hand, and with my right hand underneath I made sure the backing was smooth and straight. I removed the pins as I came to them. I used the dual feed or walking foot and sewed slow and steady. After you get out from the center, things move a little faster.
I just couldn't take all that sitting in the floor and working the pins with my hands. I wouldn't be able to sew for a couple of days after pinning one together. So I bought a longarm.