Why am I getting very different measurement results on a breadboard?
You are short-circuiting the adapter, first with the ammeter, and then with the breadboard. Both are not good for the adapter. 







simulate this circuit &ndash; Schematic created using CircuitLab

The dashed boxes represent the equivalent circuit of your supply and your ammeter; numbers are very approximate. 

On the right hand side you see the ammeter shorting out your supply and causing it to overheat and probably fail in short order. On the left hand side you see the same thing happening except most of the current is flowing through the solderless breadboard and only a small amount (the voltage drop in the conductors) is getting to the ammeter. If the conductors were perfect the meter would read zero since you've shorted the meter (as well as the adapter). 

This is bad practice and you will eventually damage something.. putting an ammeter across a voltage source can cause serious injury or death if the source is capable of a lot of current, and certainly can damage the meter or the electronics. Don't ever do it.