Are there any race tracks in north Louisiana to legally test the 0-60 MPH of my car (when I buy a car)?

Toyota avalon isn't a sporty car but to just accelerate use an interstate or a highway with traffic lights and at the red when it goes green just floor it. Just to experience it. 

Yes, it is called interstate 10.

You are just wasting time.  Millions of car owners see their vehicle as a transportation appliance, not something to get their egos bigger.

You don't NEED a track for your stated purpose.

The on ramp for the interstate is MORE THAN ADEQUATE.

Maybe "Twin City Raceway" https://www.twincityraceway.net/, which is a drag strip. I don't know if they will let the track be used just to do a timing run.

Other race tracks do exist:

https://www.racingin.com/track/louisiana.aspx

But again finding when they (or if they can be) used by the general public is a different thing.

Now I happen to like lateral G's (cornering). So I tend to look for Solo II events. These are put on by SCCA through the spring, summer and fall. These are events that allow you to race what you drive to the event. Only one car on the track at a time. So there is minimal risks to the driver. There are "drivers schools" for a small fee to SCCA members. But not required. I think the region you would want is this one:

http://www.rrrscca.org/

March 6th is the first event (Test and Tune).

You  can enter most freeways that way. Press the damned gas pedal.

You don't need a race track, just a stretch of 55-60 mph road out in the country a bit.  Turn around at a side road - go back to the stop sign, look both ways and when there are no cars pull out slowly, straighten the wheel and nail the throttle - stop when it hits 60 - no harm no foul. You will likely be disappointed in the experience overall.