Since the problem is absent warm stopped&park but appears as soon as you shift to drive, my money would be on the Transfer Case. As you fixed a 2 -> 3 shudder, the trans replacement was useful but it seems something else was disturbed in the trans swap. You eliminated the trans itself and the torque converter as a source of the issue, the remaining part common between these situations is the TC. At 250+k miles, this is quite possible. (Also my gut feeling is the TC fluid should be changed every 50k minimum but that is not part of this story.)


Reprogramming is indeed unlikely to fix the problem. But have the shop, that swapped the trans, drain the TC and inspect the fluid coming out closely. Also remove the TC actuation motor and gears, and inspect for proper functioning.

__________________
2003 4.4i black ext./black int./black headliner (kid's runabout)
2014 535d X-drive, M package, silver/black interior

Sold but not forgotten:
2009 E70 35d, black ext./black int./black headliner (sold 2021)
2006 4.8iS Le Mans blue/cream int./black headliner, SOLD in 2012 sadly...

Other hardware:
2015 Cayenne S (wife's new DD and definitely wroooommmmm)
2016 Tundra 1794 edition crew cab, luxo hauler
2005 Tundra crew cab, weekend hauler. Sold after 150k very happy miles.