Cory Monteith -- Happy & Healthy Just Weeks Before Death

Cory Monteith appeared the picture of happiness -- giggling and laughing with girlfriend Lea Michele -- this just a few weeks before he was found dead in a hotel room ... possibly of an overdose.

In this video, shot about a month ago at LAX, Cory and Lea arrive hand-in-hand looking happy and healthy. Even our photog tells the couple, "You look like a million bucks."

At the time, Cory had recently completed a month-long stint in rehab ... for admitted substance abuse issues.

As TMZ previously reported ... Cory was found dead in a Vancouver hotel room Saturday, alone, after spending Friday with friends. There are reports drugs may have played a role.