For awhile I was also experiencing roll tearing on old rolls, but not
on new ones.  The pattern was rather consistent.  I adjusted the take
up spool slightly to bring it into perfect alignment with the roll.

Also, especially on old rolls, I found it is important to add tape
to the core of the left end piece and then make sure it is pulled out
slightly from the edge of the paper.  Some people advocate the use of
O-rings for this purpose, which will work just as well.  I haven't
had a problem since.

Vincent Morgan, New York City