Does anyone know of a good book that explains Spanier-Whitehead duality (other than Adams)?
Well, one classic source is some exercises in Spanier's book on algebraic topology (alas, I don't have my copy at hand so I can't give a more precise reference, but it is towards the end).
There is also a chapter on it in
MR0273608 (42 #8486) Cohen, Joel M. Stable homotopy. Lecture Notes in Mathematics, Vol. 165 Springer-Verlag, Berlin-New York 1970 v+194 pp.
However, I have to admit that I find Adams's book very clear and beautiful. Is there a reason you don't like it?
You might find the following interesting: http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf
even though it is not quite what you asked for.