Jenelle Evans' husband, David Eason, won't be charged for the killing of their family dog ... and investigators say halfway through their findings, Evans switched up her story and said the whole thing was a PR stunt.

The Columbus County Sheriff's Office says when deputies first responded to Evans' home, she stated David took the dog outside and she heard a gunshot ... though she never saw the dog get shot. Investigators say several weeks later, during a follow-up interview, Jenelle walked her story back, saying she never even heard a gunshot and didn't know what happened to the dog.

According to investigators, Evans claimed the whole thing was simply for publicity. However, cops say Jenelle disclosed she just wanted the whole thing to go away ... so it's possible the publicity stunt excuse was just to end the investigation and protect her family.

Cops claim they did a thorough search of the property and never found a weapon nor any blood or evidence of an animal killing. Combine that with Evans saying she made the whole thing up ... and investigation wrapped.

As we reported ... the saga began May 1 when Jenelle claimed on social media she was heartbroken and distraught over the loss of her French bulldog, Nugget. Her husband, David, said he killed the dog because it bit the couple's 2-year-old daughter, Ensley.

The alleged dog killing had serious implications for Jenelle's family ... she temporarily lost custody of her three children in the fallout.

The Columbus County Sheriff's Office says it's closing the investigation and will not file charges against Jenelle for filing a false police report.