Nobody wants to lose New Orleans. It's a great city. But the problem is accurately summed up by the OP: "How much can places where millions of people live be fortified? At what cost? And who pays for it?"
The only good news is that the Port of South Louisiana is the biggest and arguably most important port in the Western Hemisphere (and that's without even taking into account the Port of New Orleans or the Port of Baton Rouge, both in the top 10). There is enormous nationwide economic and geopolitical incentive to protect the city. The bad news is that New Orleans would be sinking even without rising sea levels.
[1] https://www.logisticsmgmt.com/article/top_30_u.s._ports_big_...
Total Tonnage US:2 South Louisiana, LA, Port of 238.0
3 New York, NY and NJ 136.6
4 Corpus Christi, TX 111.2
5 Beaumont, TX 101.1
6 New Orleans, LA 92.
7 Long Beach, CA 80.7
8 Baton Rouge, LA 73.4
9 Los Angeles, CA 63.0
10 Virginia, VA, Port of 61.7
11 Lake Charles, LA 58.0
13 Mobile, AL 56.9
14 Plaquemines, LA, Port of 52.8
15 Baltimore, MD 44.2
16 Savannah, GA 41.9
17 Texas City, TX 41.3
18 Huntington - Tristate 36.8
https://www.bts.gov/content/tonnage-top-50-us-water-ports-ra...