Fain is (1) an adjective meaning glad or content to do something, and (2) an adverb meaning willingly or gladly. It’s mostly archaic. It still appears occasionally, but in most modern examples we can find, the word is used to affect an archaic tone. Feign is a verb meaning to pretend, to give a false appearance of, or to imitate.
Plainness has its peculiar temptations and vices quite as much as beauty : it is apt either to feign amiability, or, not feigning it, to show all the repulsiveness of discontent. [Middlemarch, George Eliot (1874)]
Democracy has been so retarded and jeopardized by powerful personalities, that its first instincts are fain to clip, conform, bring in stragglers, and reduce everything to a dead level. ["A Backward Glance o'er Travel'd Roads," Walt Whitman (1888)]
Dr. Bailey admitted on cross-examination that the prisoner’s memory was reasonably clear and that it is possible to fein insanity. [Montreal Gazette (1911)]
The quota of her citizens which still would fain adhere to the “traditional attitude” is very manifestly diminishing. [The Age (1940)]
But Reins refused to let that happen, Lozow said, at times feining suicide in order to maintain his attentions. [Denver Post]
In South America the man who at home calls himself an “American” is fain to modify his claim to the two continents, and describes himself as a “North American.” [New York Times (1900)]