re: Any wood furniture repair stores open on Saturdays in Baton Rouge area?

I would personally use gorilla glue or epoxy in the holes with the screws then ratchet straps to really pull the legs together while it dries.

At least two Rachet straps per chair

One around the legs and one over the seat and round the 1st set, pulling everything tight AF

While it sits overnight and dries.

ETA: If you use those Mr. Grips from Home Depot, cut them into thin strips to put 2-3 into the hole before reassembling and tightening the screws.

I’d still use the glue with them to fix the problem for good.