It's a 2005 ls sport v8 with <80k miles. I haven't serviced the transmission since I've owned the car (going to be 4 years in a few months), and I believe maintenance was minimal with the previous owner...
With the age and milage... I'm going to assume that your LS has seen mostly short trip... in town driving. That would mean that the shifter has seen a lot of use over the years.
If that is the case... I would suggest a shifter cable adjustment... and replacing the Transmission Range Sensor.
Most people call this the neutral safety switch, or reverse light switch... but it does much more since it allows the PCM and transmission to communicate with each other.
An out of adjustment shifter cable and/or worn transmission range sensor could possibly be causing your symptoms.
If I am wrong... you are only out about $40, and an hour of your time... if you do it yourself.
Jack up the car... and solidly place it on 4 jack stands !!!
The cable adjustment and sensor replacement should be done in neutral. Once in neutral... disconnect the shifter cable from the shifter arm on the transmission.
Then replace the range sensor. Take note that there are specific "line up" marks on the sensor, and that the new sensor is lined up from the factory... already in the neutral position. You could also use a paint marker and put your own "line up" marks on it... if you want to.
Once the new sensor is installed, (still in neutral)... observe the location of the end of the shifter cable... and see if it lines up exactly with the ball on the shifter arm.
If not... trace/follow the shifter cable back towards the shifter. You will see a little bracket with a white part that has "saw teeth" on it. This is where you adjust the shiffter cable. There is a little lock type piece holding both halves of the adjuster together. Removing the lock will allow you to slide the cable back and forth for adjustment.
Adjust the cable... so that the end of the cable lines up exactly with the ball on the shifter arm, and snap the cable on the ball.
Then lock the teeth at the adjustment spot... back together. Make sure you connect the electrical harness to the range sensor before lowering the car.
Then take it for a test drive and see how it functions.