Won't engage forward or reverse gears - kinda

I have a 3.0L Auto (4R44E) 2000 Ranger, 202K. It had many owners and has a long history of repairs. However it is still fairly stock. The engine was replaced with a reman 99 3.0L. Engine runs nicely, very smooth and has power.

However the trans is a different story. I will document the problem below, but I would like to point out the more we drive the truck the less the problem occurs. But all we have to do is let it sit for multiple days and the problem reappears.

Problem;
When placing it in Drive, the trans does not engage a gear. I can pull it into low gear, and same thing, does not engage. I put into reverse, however I get same thing. I have tried many combinations including letting it warm up and nothing changes. However if I rev it a "little" (I know that is not good), more like feather the throttle, it may engage slightly and the truck will move. Once I get the truck to move, it shifts fine up and down. It will engage in reverse and all seems well. But then randomly when stopped at a traffic light or at the local soft server ice cream drive through it will "slip out" of gear and will not reengage until I "feather" the throttle.

I would like to have an idea what is wrong with it before I take it to a trans shop (I currently don’t have relationship with trans shop) for diagnostics. I am worried I will get the automatic $3000 rebuild and it may be a simple fix.

Where should I be looking for the problem? How can I diagnose the problem myself? Any suggestions appreciated.