Measuring internal resistance of ammeter with ohmmeter
Does this mean the I can simply load my circuit with the ammeter

Yes, but only on that scale, and only if the applied voltage does not cause current to exceed the meter's full scale reading.
The meter may have over-current protection consisting of back-to-back diodes which conduct when the meter voltage goes over eg. 0.6 V. The resistance will then be much lower than 1k, causing high current flow that could damage the meter or the device under test.
On higher current scales the meter resistance is lower, which could also be dangerous as it is closer to a short circuit.