Both Uses of

divine

in

Much Ado About Nothing

Call me a fool; Trust not my reading nor my observations, Which with experimental seal doth warrant The tenure of my book; trust not my age, My reverence, calling, nor divinity, If this sweet lady lie not guiltless here Under some biting error.†
Scene 4.1

divinity = the state of being god-like; or of being a god

Definitions:

(1) (divine as in: to forgive is divine)
wonderful; or god-like or coming from God
(2) (divine as in: divined from tea leaves)
to predict or discover something supernaturally (as if by magic)
(3) (divine as in: divined through intuition)
to discover or guess something -- usually through intuition or reflection
(4) (meaning too rare to warrant focus)
In the time of Shakespeare, divine was sometimes used as a noun to reference a priest or a person of the church. (To remember that sense, think of the clergyman as having come from God).

Divinity typically refers to a god or to a school of religion, but on rare occasions, it refers to the name of a kind of soft white candy. To remember that sense, you might think of it as tasting divine/wonderful.