When someone's lovesick, they are broken-hearted or so distracted by love that it affects their daily life. You might be lovesick if your love took off for Bali (unless of course you have plans to take off for Bali too).
There are many different reasons for being lovesick: a lovesick boy might have a crush on a girl who barely notices him, and a lovesick teenager could be overwhelmed by feeling such strong emotions for the first time. Your grandmother's collection of old letters might include the ones she wrote as a new, lovesick wife when your grandfather was in the Army. Lovesick dates from the sixteenth century.