We are doing the opposite this summer. Flying from GSP to Oregon for five days then train to ABQ and returning on the train to GRV after a week in Albuqueque. We have left our car for weeks at a time at the GRV station with no problems when going in both directions by train. GRV is a crew change point for the RR so there are always freight crews getting on and off. Note there is a QT gas station a half dozen short blocks from the station to fill up with inexpensive gas before or after. Do lock your car and hide any valuables out of site.
We plan to Uber or call a taxi to take us to the airport. It's about 15 miles.
Parking at the station is free. Parking at the airport cheapie lot is $4/day. The nearby lots are up to $8/day. Close in is reasonable only for hourly.
Don't know if you looked at sleeper or just coach. You might consider GRV to ATL in coach then ATL to NOL in a Roomette. That will entitle you to each get Breakfast, lunch and dinner included in the cost. You can have your breakfast while the train is sitting in the Atlanta station. If you are new to Amtrak, I would strongly suggest it for the experience. Note that dinner is abbreviated - not the full menu and served a little earlier so the crew can be ready to get off the train on arrival in NOL.
Note that there is a streetcar (trolley) that runs every 30 minutes from the NOL Amtrak station to the French Quarter area. If you are a senior, fare is only $.40 with free transfers. The driver may not tell you the senior price if you don't know to ask. They try to rip off dumb senior tourists, I guess.