Police just released body cam video from the Gene Hackman death case ... and you see officers searching his New Mexico home before ultimately finding his and his wife's bodies ... with one of their live dogs lying next to Betsy Arakawa.

TMZ obtained the footage Tuesday and you see cops scouring the Santa Fe home in search of Gene and Betsy.

Police find Betsy's body first -- she's on the floor near a locked exterior door, with one of their dogs by her side ... and Gene's body is later found near the kitchen.

A different dog was found dead in a crate ... the pooch had been crated after a veterinary appointment.

The footage also shows the state of the home Gene and Betsy were living in ... and it's disorganized, to say the least.

An environmental assessment by the New Mexico Department of Public Health found Gene and Betsy's property was infested by rodents ... with dead pests and nests scattered throughout 8 detached outbuildings.

Betsy died Feb. 12 from hantavirus pulmonary syndrome ... and Gene died a few days later, around February 17 or 18, the result of heart disease and advanced Alzheimer's.