The 1841 British edition accidentally replaced the plural sicknesses with the singular sickness. The 1849 LDS edition restored the correct plural. As discussed under Alma 9:22, either grammatical number is possible for the word sicknesses(es), although the Book of Mormon text prefers the plural. For each case of sickness(es), we follow the reading of the earliest textual sources.
Summary: Maintain in 3 Nephi 7:22 the plural sicknesses, the reading of both 𝓟 and the 1830 edition (the earliest extant sources).