The distance they travel is, of course, speculative. Sorenson suggests that they would make no more than 11 miles per day, based on comparative estimates for similar situations (Sorenson, John L. Mormon's Map. FARMS, 2000 p. 56). This gives us approximately 88 miles away from Mormon, which was another three days or 15 to 40 miles away from Lehi-Nephi (Sorenson 2000, p. 56). Since they were fleeing Noah's army, they certainly would not have gone back towards Lehi-Nephi (and therefore Shilom). The route appears to be in the general direction of Zarahemla, as they will go from this new location to Zarahemla without doubling back on their previous travels.
They come to a place that is explicitly a location of "pure water," a phrase we have also seen in Mosiah 18:5 as a descriptor of the waters of Mormon. As noted in the comments on that verse, this may relate to a Mesoamerican conception of the purity of waters from earthly sources (Sorenson, John L. An Ancient American Setting for the Book of Mormon. FARMS1985, pp. 176-9. See the comments on Mosiah 18:5 for a further elaboration on this concept).
Textual: The printer's manuscript and 1830 edition of the Book of Mormon both begin this passage with "And it came to pass that they fled…." This was deleted from 1837 to the present (Book of Mormon Critical Text. FARMS 1987. 2:473).