The sexual meaning dominates the modern use of the word, and was indeed present from the time when it first came into English, during the 1590s.
Shakespeare is actually the first recorded user, but he employs it in a more general sense, as an intensifier of disgust - "repulsive, offensive".
There are just three quotations. Prince Hal calls Falstaff a "whoreson, obscene, greasy tallow-catch" (Henry IV Part 1, II.iv.224). The Bishop of Carlisle talks of Richard's overthrow by Bolingbroke as "so heinous, black, obscene a deed" (Richard II, IV.i.131). And the King in Love's Labour's Lost reads Don Armado's letter describing an "obscene and most preposterous event" (I.i.236) - referring to no more than Costard's meeting with Jacquenetta within the court precinct, from which women have been banned.
In modern English there are signs of a return to this intensifying sense.
When we say "he was paid an obscene amount of money", we mean "disgusting", but without the sexual connotation.