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TMZ Sports has obtained the Lawrence Taylor DUI video from the night he crashed into a cop car on a Florida highway ... and it shows the NFL legend was wobbly and confused before having a medical emergency while in handcuffs.

Taylor was detained by cops in Sept. in Palm Beach after he slammed his Bentley into a police cruiser and a trailer around 5:30 PM.

Officers put Taylor through multiple field sobriety tests -- including walk the line and balance on one foot. You can see in the video ... it did not go well for LT.

Taylor was arrested on the spot for DUI -- but while in custody, he started to breath heavy and wail in pain. Cops called an ambulance and LT was transported to a nearby hospital for treatment.