I've read about this concept in the Riverworld series.
It doesn't end well for them.
Random DNA sprayed around a crime scene would just seem weird. Wouldn't they also have to make it seem real? IE fake blood, saliva, whatever, in the same places it would be found in a crime scene?
The meat is rotten, but the booze is holding out. Computer translation of "The spirit is willing, but the flesh is weak."