Kind of an odd shaped parcel and oddly oriented to directions....
So the flower bed is on the north side of the house and is pretty shady? The open area you mentioned that is low lying also gets the best sun? The deck faces west-ish and gets afternoon sun, but is shaded by the trees and shrubs in back?
So here's just some thoughts of a plan off the top of my head. Obviously there's nothing "right" about this, pretty much personal preferences. It is upside down now, but click on it for right side up and better view.
So what I was trying for was putting the veggie gardens where the best sun is. If you build raised beds, that solves a lot of the drainage problems. You just may need to build them taller than you might otherwise, like maybe 18" or so. I gave you the configuration of raised beds I have in one section of my yard
It's 144 sq ft or so of raised beds (four 4x8' beds, with a 4x4 in the middle), which is plenty to start with if you don't have a lot of veggie growing experience. The playset is close to the house and near deck and patio for ease in watching the children and getting to them if needed. The fire pit with seating is tucked away in the most remote/ secluded area. You can have lots of flowers and flowering shrubs along the fence line, but especially for the S/E fence line, keep them low so you don't shade out those veggies. Your existing flower bed is tiny; it could be expanded to be a lovely shade garden.
Hopefully others will chime in with their thoughts!