Lustful is the adjective corresponding to lust in the sense related to sexual craving, so it means full of sexual craving. But sometimes lustful means just full of craving, without necessarily involving sexual feelings. Dictionaries list lusty as a variant of lustful, but its primary definition in modern English is full of vigor or having robust health.
Both adjectives have long histories involving many now obsolete definitions. Lust, along with lustful, is from Old English, where lust meant pleasure, and the OED lists one instance of lustful (spelled lustfull) from as far back as the ninth century. Lusty is a little newer. Two of its earliest known uses are found in the works of Chaucer, writing in the late 14th century.
Examples
Here … sat the lusty coalheavers, quaffing large draughts of Barclay’s best, and puffing forth volumes of smoke. [Sketches by Boz, by Charles Dickens (1837)]
An odor as of breakfast came stealing through the wood; the Paladin unconsciously inflated his nostrils in lustful response. [Personal Recollections of Joan of Arc, by Mark Twain (1896)]
Some such plan had lain as a seed in his mind for many years; and now that he had decided, in a flash, to give up his profession, the seed grew in the space of twenty minutes both tall and lusty. [Night and Day, by Virginia Woolf (1919)]
Jones remarked with slow, lustful approval her firm, free carriage. [Soldiers’ Pay, by William Falkner (1926)]
His godmother, Margaret Clifford, remembers him as a “nice, large, lusty baby who liked to play with a ball in his crib.” [New York Times (1962)]
Through history, many lustful victims have been entrapped by the lure of sex but the natural world has revealed a truly bizarre example of the honeytrap. [BBC (2000)]