Q:

How can I edge slide to an accurate value on one axis? 

I want to slide this edge to 38.4cm on the Z axis.  The only way I know how to slide edges is double tapping G and sliding it with the mouse, which is not accurate enough for my needs. 

Is there a way for me to type in a specific global value without messing up the right side of the mesh? 

A:

Sorry, previously added a completely wrong answer...
After some struggling I agree that is a shame that you can't set a precise sliding end position (at least on global axes) or at least snap to some other vertex or edge... this should be fixed, imho (although it is not currently considered a bug... https://developer.blender.org/T41346). 
I have another workaround, not very much elegant but at least you could try this, too, at least this is what I tried:
First I duplicated this edge:

then, I selected those edges, whose Z value was 0.00, and grabbed (not slided) them to a precise value (0.2 in my example)

of course ending with this wrong result, though you can see the duplicated edge I created before, as a "reference" that keeps the correct slope angle...

then, tried to correct the wrong vertex position (only one in this case), after activating snap to edges

I then grabbed (locking on X axis) until the vertex snapped to the "reference" edge)

Finally, I had to delete the "reference" slope edge

here you see the "real" edge selected after removing the "reference" one

That's it, I hope at least this gives you an alternative...

