The 1874 RLDS edition replaced the subordinate conjunction that with the coordinating and, perhaps unintentionally. The 1908 RLDS edition restored the original that. Here we have a resultive relationship between the two that-clauses: the Nephites slaughtered the Amlicites, with the result that the remaining Amlicites started to flee. Similar examples can be found in the text where pairs of that-clauses are asyndetically conjoined, as in the following example where the second pronoun they has a different referent from the first they (just as here in Alma 2:18):
Thus the critical text will maintain in Alma 2:18 the two conjoined that-clauses for which there is no connecting and.
Summary: Maintain the subordinate conjunction that in Alma 2:18; sequences of asyndetically conjoined that-clauses can be found elsewhere in the text.