While I don't have a specific reference for someone having an "earworm" in the 19th c. or earlier, almost certainly it happened. Musicians were often expected to be able to improvise, either pulling a melody out of their heads, or taking a popular one and doing variations on it. Schubert was famous for his ability to do this, but in earlier centuries it was even more commonly expected. As a result, theme-and-variation was a very common musical form, and lots of them were printed. Of course, the catchier the tune, the more popular it would be as a theme and so the more often variations on it would be printed. One good example would be the settings of Jacob van Eyck, a blind Dutch 17th c. recorder player whose Variations are still played . Here's his setting of an English song, "When Daphne from Fair Phoebus Did Fly". You can find other printed settings of it, for other instruments, where the tune was just called "Daphne". Other names pop up constantly in this period, like Packington's Pound, Amaryllis, and Greensleeves.

A good tune would also tend to be used for other songs. If you went to a fair, likely someone would be selling broadside ballads, songs with topical or amusing lyrics. Rather than compose and print the music for a new melody, the balladeers would set them to something already popular, and so while a lute-player might be showing off her variations on Packington's Pound, a ballad seller in the market nearby might be hawking a number of new songs set to it as well.

For something classical, maybe the most famous would be the 1761 French song "ah, vous direz-je, maman", for which even Mozart composed some nice variations. It's still being sung, as " Twinkle Twinkle Little Star".