Yes..first safely raise vehicle on drivers side and secure. Safety first! Parking brake engaged etc.
BTW...I note your transmissionis not in drive when the indicator states so. Before following the rest of my directions... you might want to make sure the shift linkage is adjusted. If it isn't that probably explains the issue. BTW your symptoms suggest this, not a neutral switch.
To adjust shift linkage, first have a helper run the gears so you can watch the linkage.
To make sure your linkage is in park with the shifter in park, remove the linkage rod from the column to the transmission and then see if the linkage pops into park all the way. Many times the park pawl will engage but linkage will not close the electrical circuit in the switch.
Make sure there is no mud, dirt, wasps nest, dead animal or anything jamming the linkage. It can be adjusted by either a clamp bolt on the transmision shaft link, or on the shifter rod itself.
Once you've determined this isn't the problem....
Get a drain pan, you'l need one/ Also some tranny fluid.
Remove positive battery cable.
Locate the neutral safety switch on drivers side of transmission.
Remove electrical plug. Note: To ensure diagnosis is correct, carefully inspect plug and its wiring for several inches to make sure the harness is not damaged be exhaust heat or physical means.
Place drain pan under tansmission. Have new switch ready to install, because when it comes out so will ATF!
Remove and replace switch quickly (with appropriate wrench or socket) to minimize fluid loss.
Rest of installation is reverse of removal. Top off transmission as needed.