NFL's Will Smith Barely Drank at Restaurant Before Shooting

Will Smith did NOT appear drunk when he left a New Orleans restaurant ... moments before he was shot to death -- this according to restaurant management.

TMZ Sports spoke to employees at Sake Cafe Uptown ... the last place Smith was seen before the shooting. The incidents with Cardell Hayes reportedly began after he left the eatery.

There are theories that Smith was sloshed at dinner ... and was driving impaired when he got into it with Hayes.

But management at Sake Cafe Uptown say they're not buying it -- insisting Smith only had 1 glass of wine during the roughly 75 minutes he was at the restaurant.

"He got there kinda late ... didn't drink a lot at all ... no cocktails, and I can only remember seeing one glass of wine," one staffer tells TMZ Sports.