2004 Honda pilot. Car only moves a little in drive and pops into false neutral?

Hello, all signs point to internal mechanical concerns with the transmission. Black fluid going out is never good. A lack of engagement in Drive is never good. Electronic testing is needed to determine if you have a possible electrical concern contributing to the engagement issue. There's a small chance you may have some kind of concern with a pressure sensor or range sensor. Testing will confirm if all seems OK or if an area needs troubleshooting. You could try another fluid change, in case you still have "black" fluid present in the transmission; perhaps it could help. But usually when engagement is a problem, it will take hard part replacement to resolve the concern.