The verses in Matthew are:
25 Agree with thine adversary quickly, whiles thou art in the way with him; lest at any time the adversary deliver thee to the judge, and the judge deliver thee to the officer, and thou be cast into prison.
26 Verily I say unto thee, Thou shalt by no means come out thence, till thou hast paid the uttermost farthing.
The 3 Nephi teaching adapts to the New World context by removing the layers of authorities that were part of Jesus’s world. The Old World farthing becomes a New World senine (see Alma 11:3–8 for the definition of a senine).
To be “in the way with him” means to be on the same road or path. Thus, differences might arise, but they should be settled amicably at the time, and not accelerated to the point that legal authorities would become involved. This admonition is arguably more relevant to what we know of Jesus’s Jerusalem than to what would be present in Bountiful at the time of the sermon.