Jameis Winston Accuser -- BEGGING FLORIDA GOVERNOR ... Investigate Tallahassee Police!!!

Jameis Winston's accuser believes the Tallahassee PD completely BOTCHED her case ... and now she's begging the FL governor to launch an internal investigation, TMZ Sports has learned.

The accuser's attorney, Patricia Carroll, tells us ... her client fired off several letters to various officials, including the FL Attorney General, the Federal Dept. of Law Enforcement and Gov. Rick Scott asking for them to help make sure the PD is properly investigated.

As we previously reported, Carroll says the PD bungled the case from the start -- failing to interview witnesses, not testing blood for rape drugs, sitting on the case for 11 months and more.

In fact, Carroll says the accuser has recently been bombarded with letters from other alleged rape victims detailing similar experiences with the Tallahassee PD.

The accuser is going to have a tough battle getting Gov. Scott on board -- considering he recently shot down her prior request for an investigation.

But the accuser hopes that with other women coming forward with similar stories, the Gov. will change his mind.