Is there a way (a library) with which one can give the (contour) lines of AWT-2D-objects a "hand-drawn" look (somehow shaky: not exactly following the "official" path, not having perfect line contours)

(The official path should be followed almost perfectly, with just some "random" noise. The line contours should be almost perfect, with just some "random" noise.)

Once I tried to do something similar with jhlabs. Please take a look at the WobbleStroke sample at: http://www.jhlabs.com/java/java2d/strokes/
In general - you have to implement the java.awt.Stroke class. So better search for appropriate implementations.
Here is a good sample of a brush stroke as well: http://javagraphics.blogspot.com/2007/04/strokes-brush-stroke.html