I was in the same boat, putting was my worst part of my game for 35 years. First I studied putting, by reading several books on it, then practiced what they taught. Second I got fitted for an 'Arm Lock' putter, because I had a hard time keeping the putter face from changing alignment during the stroke with standard putters. Third is I changed my mind set from, "I hope this stops close if it does not go in", to "I am going to make this"! This helped to reduce or eliminate the fear of missing a putt and having a 4 to 5 footer left. Funny thing is, even if I missed, the putt was still usually within tap in range.
Because of my poor putting in the past, during best ball tournaments I was always putting first in order to give the other players the line. After the three changes above, I was still putting first, but was holing almost everything (including 20 foot breaking putts).
Putting, like most golf, is a game of confidence, and sometimes you have to fake it until you make it!