I have approx 20 layers in illustrator (a world map with layers for capital city names, small city names, rivers, oceans, etc). I want to increase the size of this map, so I need to increase the size of every layer without distortion. I have tried to Select all - group - transform, but all layers disappear off the artboard and are bigger than the dimensions I typed into W x H boxes. Any advice appreciated. Thanks!
What you describe should work fine (although you don't need Shift if the chain icon is active).
With the chain active, you do not want to enter both width and height. You only want to enter 1 dimension. The chain icon tells Illustrator to automatically calculate the other dimension proportionally.
So, you would enter a width and hit Enter. That's all.
If you want to non-proportionally scale everything, unclick the chain icon and then enter a width and height.
It sounds as if you are ticking the chain icon (keep proportional) then entering a width, then entering a height. Since the height is the last dimension entered, Illustrator is making the width proportional to that height.