Board Game Arena

Am I missing something? New to the game and I read that you draw 5 objectives to start with and keep 2.

But we've started playing and I haven't had to get rid of any and we all still seem to have 5....

You are referencing the 1st edition rules. The second edition changed the rules to be as follows:

Rule Detail (a doffing of the top hat to Martin Butcher)
► Setup: For a standard game, players are dealt five OC, keeping them all. For an extended game, players are dealt four OC, keeping them all.
► During the Game: At the end of each courtship, players discard one OC.
► Round Track: During rounds featuring an OC draw, players draw two OC, keeping both.

In standard play, players acquire a total of seven OC, discarding four, scoring three. In extended play, players acquire eight OC, discarding four, scoring four.