I mean, on one hand, I could see the templar red lyrium prob starting in Kirkwall with that Meredith statue in courtyard corrupting people. Like we've discussed a billion times. But... Cullen doesn't seem to know all that much about templars taking it, which does suggest either Hawke knows more about his templars than he does (unlikely, unless he was still in his oblivious phase
), or he left earlier than all that.
He couldn't have left before Hawke, because his letter to Mia and his conversations with IQ place Cassandra's offer to join the Inquisition as his final deciding factor in leaving the Templars, and Cassandra only arrived in Kirkwall three years after the Anders incident.
It doesn't seem very complicated, though - the Kight-Commander is very busy, and not all-knowing. The Templars did all kinds of stuf without Meredith's or Cullen's knowlege before (he does tell you he thought everything was normal at the Circle but he should have looked harder into it). Thrask was even able to colaborate with the mage underground and start his own rebel cell among the Templars, right under Meredith's eye. After the destruction of the Chantry and the rebellion in the Circle, Cullen would be stretching his attention everywhere, the survivor searches and reconstruction efforts in the city, trying to run the Circle with what's left of his men and the few surviving mages, working with Aveline and Hawke to keep the peace etc, and Templars messing with Red Lyrium in secret could easily have escaped him while they didn't yet develop the visible symptons.
Those Templars could have involved Hawke in dealings or blackmail (she does have seven years of dirty laundry around the city), without using the Order's name, making it even more unlikely that Cullen would find out unless Hawke told him about it. She might have chosen to leave some time around this, for... whatever reasons, leaving Cullen with a growing problem he didn't know about, or even did know, but couldn't combat on his own. At the time Hawke leaves, it's possible Cullen did already know about the Red Lyrium problem, but his leadership among the remanining Templars in Kirkwall is already slipping and there's not much he can do, contributing to his frustrations and making him consider quitting, and that's when Cassandra comes in with her offer. (at least that's one way to make him quitting make sense)