From the Zone, which is the go-to resource for all things SUP:

Paddling thru a whitewater standing up is fun.  I never go to the water to hold my board, not even turtle dive.  There are 4 techniques I use to punch whitewater and holding on to the tail of my board is not one of them. 
Wave to big   station keep wait inside for lull and then haul ash.  If to late dive away from your board.
Caught near impact zone    stand up and take it like a SUPer.  If to big dive off to side of board. I make some big whitewaters by trying.  I surf a beach break so no benefit of a channel.  The biggest surf I paddle out in is around 8′, anything bigger without a long duration in the swell is a beating asking for injury.

There is no time to grab a leash until you are being drug by the board.  I also keep my distance from others when paddling out and plan to take the impact zone for a surfer riding the wave in.  So I am avoiding close proximity to others especially in front of me or behind me.  This reduces your chances of a collision.
White water medium big I can sink the tail and blast over the top in a prone position.  If already in the water (due to a trashin) I will put the paddle on the deck and lift the nose over the whitewater hanging on as I go thru. Duck dive the whitewater by sinking the nose and forcefully pushing thru then recovering on the back side. Put your paddle under your body as this is from prone paddling position.   These techniques come directly out of the surfers manual.  I guess you could turtle dive(flip board over) but that never has a clean exit.  Having a surfing background is very beneficial for the surf zone.  Experience is knowledge.
The only time I grab my leash is when I am recovering my board and I pull on it to reel it in.  Even paddling out how you gonna wrap your leash around the paddle and paddle.
Your focus should be on success not recovery.

Wrapping your leash around your paddle with big wave impacts.