Two things.
1) I noticed, for me, that when I had a full-offset Anser-style putter (Newport 2 for me) I had a harder time taking it back on the proper path than I did with a half-offset Anser-style putter (Mizuno BC1 for me - Scotty makes the 1.5 in half-offset, I believe). I was told this had something to do with my being right-eye dominant.
2) I also noticed that my putting improved when I improved my putting setup, alignment, and grip. The grip, above all else, is key in putting. Interestingly, I'm still more comfortable with half-offset putters, but I don't jerk full-offset putters around anymore.
I'd pick up The Art of Putting by Stan Utley and take it to heart. It will definitely help you. As a secondary solution, check out different putters until you find a comfortable one.
Keep in mind that it's okay for the putter to move a
little
closer to you on the back swing, because that's the first half of a "gated" or arcing swing path (nothing wrong with that). It just shouldn't be a dramatic inside move. So don't try to make a perfect straight line, but instead try to make a consistent slightly arcing path.
Straight Back Straight Through ("SBST") putting is a total joke IMO, both because almost all adamant supporters of SBST don't even putt SBST and because it requires path correction with the wrists and/or hands.