different current readings with a multimeter between the 200 mA and the 10 A slots
Measuring the cheap meter on my desk, the 10A range uses a 0.1 ohm shunt resistor and the 400 mA range uses a 2.5 ohm shunt resistor and the 4mA range uses a 100 ohm shunt resistor.

100 mA through 0.1 ohms is 10 mV drop and 100 mA though 2.5 ohms is 250 mV drop. So depending on the impedance of your circuit, the current could be lower in the lower range just because of the increased series resistance.

You can measure the voltage drop across the ammeter with another voltmeter.

When you can't tolerate voltage drop, try a feedback ammeter: http://www.keithley.com/data?asset=6169