When people say that someone is overly sensitive, they mean that this person feels emotions too strongly and takes things very personally. The word "overly" is used to describe when something happens to a great degree, more than is normal or necessary. For example, if a recipe calls for a bit of sugar but you add much more, the cake will be overly sweet. If someone is overly critical, they tend to judge others harshly. In general, "overly" helps express that something has gone beyond what's reasonable or acceptable.
How common is "overly"?
Word overly is considered uncommon in modern English. It has a balanced usage among all categories: speech, fiction, newspapers and academic texts.
Definitions
adverb
(sometimes proscribed) To an excessive degree.
Example: Parents can be overly protective of their children.
(usually negative) To a high degree; very.
Example: I'm not overly enthusiastic about meeting him.
(obsolete) Superficially.
Example: 1566, Thomas Blundeville, The Fower Chiefyst Offices Belongyng to Horsemanshippe, London, “The true Arte of Paring, and shooyng all maner of Houes together […] ,” Chapter 5,
[…] let him not touche the quarters nor the heeles at al, vnlesse it be to make the seat of the shoe playne, & let that be done so superficially or ouerly as maye be, so shall the houes remayne alwayes strong.
(obsolete) Carelessly, without due attention.
(obsolete) With a sense of superiority, haughtily.