I don't quite get what you mean by "distinct consecutive transversals", can you explain more clearly how the pre-coloring is done?


One question that I can answer: 
For the case $n=8$, with the precoloring you describe the completion you give is indeed **unique**. I checked by writing the corresponding boolean program and let a solver enumerate all solutions: there is only one. 

[![Colored C_8][1]][1]


  [1]: https://i.sstatic.net/BZYli.png