ClearAll[RosettaShortestCommonSuperSequence]
RosettaShortestCommonSuperSequence[aa_String, bb_String] :=
 Module[{lcs, scs, a = aa, b = bb},
  lcs = LongestCommonSubsequence[aa, bb];
  scs = "";
  While[StringLength[lcs] > 0,
   If[StringTake[a, 1] == StringTake[lcs, 1] \[And] StringTake[b, 1] == StringTake[lcs, 1],
    scs = StringJoin[scs, StringTake[lcs, 1]];
    lcs = StringDrop[lcs, 1];
    a = StringDrop[a, 1];
    b = StringDrop[b, 1];
    ,
    If[StringTake[a, 1] == StringTake[lcs, 1],
     scs = StringJoin[scs, StringTake[b, 1]];
     b = StringDrop[b, 1];
     ,
     scs = StringJoin[scs, StringTake[a, 1]];
     a = StringDrop[a, 1];
     ]
    ]
   ];
  StringJoin[scs, a, b]
  ]
RosettaShortestCommonSuperSequence["abcbdab", "bdcaba"]
RosettaShortestCommonSuperSequence["WEASELS", "WARDANCE"]