OK, final update (I hope!!) I have been driving it around with the D and R indicator dash lights on at the same time while the transmission in Drive for the past week. I took apart the NSS AGAIN (third time) and found nothing wrong with the copper parts inside or the connector plug. No change. I took apart the dash and looked at all the plugs and printed circuit boards for the gauges, again no change.
Next I checked the trailer light harness under the rear bumper and found it was corroded badly. I opened up the area behind the rear wheels under the sliding glass windows on both sides and checked all of the electrical plugs, all perfectly clean with no signs of water leaks. Since I don't tow anything, I unplugged the trailer light converter thingy glued to the pass. side inner quarter to take the corroded trailer connector out of the equation. Still no change, reverse lights and the D and R indicator on when in Drive. I noted that although the reverse lights were on while in Drive, they were not as bright as if in Reverse. Same with the R indicator, would come on while in Drive but not as bright.
Not knowing where to turn next, I took a chance and ordered a new Toyota NSS/ gear indicator switch that is mounted to the gear selector on the transmission. I installed it today and everything is working as it should. All indicator lights work properly, and the Reverse light only come on while in Reverse! Problem solved. Not happy about the $150 NSS but at least all of my time spent looking for the problem was free. Thanks everyone who pitched in with suggestions.