We have a 2003 Explore xlt with 40K, it shudders when accelerating and making a moderately sharp turn. It feel similar to being in 4wd, but not quite as extreme. Took it to a mechanic, he said everything looked tight and fluids are full. Trying to avoid dealer.

Any suggestions ?

You need to speak to Ford dealer about adding a special additive to the differential - should clear problem.

Did the tech check the CV joints. I would take it to a ford tech. It's s new truck and regular mechanics don't have the information to diag them. You could be locked into 4x4. maybe your shift module is not working correctly.You can have bad bearings. As for the Friction Modifier its for the limited slip. It helps eliminate noise from the hypoid gears.
Give some more symptoms. Have you jacked it up and checked the front end for play? Inspect the CV joints.