Make sets disjoint without emptying them
Jelly, 8 bytes

Œp⁼Q$ÐfḢ


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Explanation

Œp⁼Q$ÐfḢ  Main Link
Œp        Cartesian Product of the elements; all possible lists
     Ðf   Filter; keep elements that are
  ⁼Q$     Equal to themselves uniquified
      Ḣ   Take the first one


Extremely inefficient. Asymptotic to ϴ(n^(n+1)) according to Misha Lavrov; I think it's right.