I'm using the cheapest stuff from Runnings, green twine but can't remember the name.   I had all kinds of trouble last year and no trouble this year.  Traded balers.  Went from a NH to a JD and love the difference.  On every single bale the twine starts, even at the beginning of the ball.  So, your troubles may not be with the twine.

Of course, you may be flamed from SOMEONE here because you, like me, are still using 100 year old technology.  Seems the world revolves around netwrap and Heaven help you if you still use sisle.