The u-joint is not servicable as most are. The driveshaft shops like the one Jim suggests, will put in an entirely new joint.

To answer your question about inspecting the joint, both axes should rotate smoothly. Most likely you will have a "notch" or detent as you flex the joint in both axes. This is what the relocation of the carrier will hide.

Since that didn't work, you should inspect the flex disks and the carrier bearing including its rubber mount. The front flex disk fails more often because it is subjected to lots of heat. The rear flex disk deals with no heat, so usually is okay.

Also when you remove the shaft to inspect it is common for it to slip apart into two pieces. There are marks that will show you how to realign. There is a raised knot on one piece that aligns with two raised knots on the other piece.

Good luck,