I hooked up a pressure gauge set from Harbor Freight to the point where the oil pressure sensor goes and ran the engine. I found the oil pressure was great cold, warmed up, low rpm, and high rpm. I did notice the connector was easier to remove from the oil pressure sensor than those VW connectors usually are. I found a post on VW Vortex for another low oil pressure issue where someone shared a wiring diagram for the circuit. They also mentioned the oil pressure sensor (with its single lead) provides a resistance to ground (which it gets thru its body to the block). This resistance to ground is measured by the instrument cluster module. The instrument cluster module does not show low oil pressure at idle with an open circuit but does when the engine is revved up to about 2000 rpm. This had me thinking about how easy the lead was to disconnect. I think it was just not plugged in to the replacement OEM I put in earlier. I put the new OEM sensor back in and made sure the connector was all the way on. I tested it and no more oil pressure light! Hooray. I took it for a test run and no more Oil Pressure light.
So I screwed up and didn't plug the connector back on all the way when I replaced it. I hope my mistake helps someone out.