Dog the Bounty Hunter Not Engaged Despite TV Proposal to Moon Angell

Dog the Bounty Hunter -- who's been reeling after the death of his wife -- has a new woman he wants to marry, or at least he did for a moment while he was on 'Dr. Oz.'

The other shock is the alleged bride-to-be is one of Beth Chapman's friends ... Moon Angell. Dog's been rumored to be dating her for a few weeks, pissing off his daughters, who have claimed he's lying about the relationship. It's caused a rift in the Chapman family.

Regardless, Dog and Moon -- who was once Beth's assistant along with her BFF -- appeared on "The Dr. Oz Show" together ... and Dog shocked everyone by proposing to her.

Sources connected to the show tell TMZ ... he also told Moon he loved her and revealed he was talking to God in the bathroom during a break, and God told him he should propose.

However, sources close to the family tell us ... the TV proposal, recorded last week, was NOT legit -- he was overly emotional and got caught up in the moment. We're told he did not give her a ring during the taping, and they are not engaged.

It certainly didn't play out that way on the 'Dr. Oz' set though. People involved in the production tell us everyone in the studio thought Dog meant it. You can see the full interview and judge for yourself Monday on Dr. Oz's show.

Our sources say he's grown close to Moon -- who was maid of honor at his and Beth's wedding -- because she's helped him through the grieving process, which she's going through too.