"slack with" is correct and usable in written English.
You can use it to refer to someone who is not very attentive or diligent in doing something. For example: "He has been very slack with his studies lately."
The man's face goes slack with disbelief.
Liu's face looked slack with fatigue.
The face of Aoyama (Ryo Ishibashi) is slack with sadness.
David Wright has not been able to pick up the slack with Beltran gone.
They get a lot of slack with regard to the violence.
I let go and look at her again, my face slack with shock: we both laugh.
A day later, Mr. Myagchenkov's face was still slack with shock from what he had seen.
I've been eating and sleeping regularly, but I've been a bit slack with the 6 30am exercise.
Now, more than a decade later, MoMA is picking up the slack with a survey show of new art called "Soundings: A Contemporary Score," which opens Saturday.
Shots of the crowd catch faces gone doughy, slack with elation, as if they have just born witness to a true, contained miracle.
My recovery has been amazing (though my wife would argue that weekend mornings are still tough — she picks up the slack with our two kids).
