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TMZ Sports has obtained the dash cam footage from Austin Seferian-Jenkins' DUI arrest -- and he ALMOST got off scot free ... until he started running his drunken mouth.

The ex-Bucs tight end was pulled over for an improper lane change after cutting off a cop -- and moments after the stop, the officer was going to let ASJ off with a warning.

Then Austin started talking ... and slurring ... and the cop quickly realized the 23-year-old was wasted.

Austin begged the cop to let him go to his QB's house to watch 'American Horror Story' ... but the cop shot that down.

Once Austin was arrested, he told the cop he needed to take a "huge s***" -- the cop begged him not to do it in his brand new squad car.