Most noticeable when truck is heavily loaded (full of bricks or full of wood).

Somewhat less noticeable when not loaded, or when put into 4L.

Somewhat told me there is a screw that can adjust it (he had 97 1500 manual), but I wouldn't know where to start, nor do I even get the mechanics of the judder.. perhaps plates not well aligned?

More gas seems to make it worse.

Also, when put in low gear seems to not roll/slow down even when in neutral.

Truck has 126K miles. Clutch seems OK beyond first gear.. no grinding, slippage, etc.

 

That sounds like the rear motor mount, which isn't on the motor, it's on the transmission. Make sure that it is tight, and check your front mounts at the same time. Open the hood, stand to the side, and have someone put it in gear....does teh engine jump? I'll bet that although you say it is ok in the upper gears, you can feel driveline slop sometimes between gears or going from uphill to downhill.
There are two other problems I have had on Chevy trucks, both much less common than a loose/broken motor mount, which can happen on anything and has those symptoms.
(a) Check the flange bolts that fasten the transfer case to the tranny. Very often this will show itself as a leak.
(b) I have seen the pads on the axle that that leaf springs rest on bend so that the axle flexes when you put it in gear. The U-bolts won't tighten because they were old & rusty; once apart the pads had to be straightened and new U-bolts put on & tightened like nobody's business.

 

I would expect always some movement of the engine when it's put into gear.
What sort of movement would be bad?

Can the mounts be checked from under the truck?