+1 on ribeye. Lots of marbling and the tender cap piece...yum!
Give sous vide a whirl. Especially if you're cooking 3lb pieces. Uses science to give you the perfect inside and leaves you to do Art to get the perfect outside. Time consuming but if you have the time, you can't get a better result.
I find this really old post on a UK forum. The guys symptoms are identical to mine. It looks like he ended up doing something like the pause drill.