The current going through a circuit is dependent upon the total resistance (equivalent resistance) of the circuit.
V = I x Req
By adding an ammeter of very large resistance, you would increase Req, decreasing the current coming from the voltage source. So the measured current would be less.
You are correct about elements in series having the same current going through them. But that doesn't mean the current won't change if you add another resistor. The current will decrease for the entire circuit. So, let's say you add another resistor, and the current in the circuit decreases from 3A to 2A. Now all elements in series would have 2A going through them. This is less current than if you didn't add the resistor.
Hope that made sense.