To wear out, to vanquish.
a. 1848. Either flax out your opponent, or give nature special fits in the undertaking.Dow, Jun., Patent Sermons, i. 54.
1866. I think to flax for to beat [is American].Lowell, Introd. to Biglow Papers. (N.E.D.)
1903. Dialect Notes, ii. 351, quotes flaxed out as an Ohio term for worn out.