Old English sorg "grief, regret, trouble, care, pain, anxiety," from Proto-Germanic *sorg- (source also of Old Saxon sorga, Old Norse sorg, Middle Dutch sorghe, Dutch zorg, Old High German soraga, German sorge, Gothic saurga), perhaps from PIE *swergh- "to worry, be sick" (source also of Sanskrit surksati "cares for," Lithuanian sergu, sirgti "to be sick," Old Church Slavonic sraga "sickness," Old Irish serg "sickness"). Not connected etymologically with sore (adj.) or sorry.
"to pretend illness to escape duty," 1820, from French malingrer "to suffer," a slang word that probably also at one time meant "pretend to be ill," from malingre "ailing, sickly" (13c.), which is of uncertain origin, possibly a blend of mingre "sickly, miserable" and malade "ill." Mingre is itself a blend of maigre "meager" (see meager) + haingre "sick, haggard," which is possibly from Germanic (compare Middle High German hager "thin").
The sense evolution in French would be through the notion of beggars who feigned to be sick or exhibited sham sores to excite compassion. Malingerer is attested from 1761, in a translation of de Saxe; malingering as a verbal noun is attested from 1778. Related: Malingered.