Doug,
I am like you, but went to a Pelz school (one of many available) and recently got a putting training device from Canada (also similar to others with line up aids) I was on the putting green at Colbert Hills and making everything from 10 feet and in, and 90% of 20 footers with that thing guiding me.
While I didn't do quite as well going out on the course, I did break 80 on a tough course, in part thanks to putting.
Short version: It takes some lessons and work, but you can improve your putting.
RE:Feel, Pelz and others (even Tiger Woods paces off his putts) recommend knowing your distance per inch of backswing. For me, its about one putter width back for every 3 feet with an even swing. Of course, you have to adjust for late in the day, up hill and down hill putts, etc.
Short version: Try to take feel out of the equation. Go with mechanics!