When New Orleans was "established" it wasn't below sea level. A
lot of the present day city was built below sea level. Some engineer
had the bright idea of pumping it out to extend the size of the city.
Yes, New Orleans has sunk over the past 100 or so years, along with
the levees.
Check out this USGS document:

http://dels.nas.edu/dr/docs/burkett.pdf

I see this as a college student type document to try to explain what might
happen with predicted change in patterns caused by global warming etc. Ive
seen all sorts of "expert" predictions and possible results many times and
they all seemed to have valid background theory but in reality.. like most
things on paper looked good but with no REAL SOLID proof figures, statistics
and charts can be made to say anything. The figure of 9mm subsidence per
year equates to 90cm over 100 years, or 3ft. Naff all really. I think its
more along your suggested second part where you say they pumped out the
water to expand the city then expanding it under sea level rather than the
whole place subsiding under sea level since being made. . I will do a bit
more investigating on this as its caught my imagination now and it might be
interesting to study it.
Many thanks,
Graham