To zero a voltage measurement, you short the leads together, ensuring zero potential between them at the contact point, and then you zero the reading. If the leads are left open, then you can't assume they are at the same potential.
To zero a current measurement, you open the circuit. That is how you ensure zero current flows. If you short the leads, you provide a path for current, and you can't assume there is zero current. Due to the low resistance of the current path through the instrument, even a small thermoelectric potential can induce a measurable current. You can't assume that this same current will flow through a series-connected load, since the load will necessarily have a much higher resistance than the shunt resistance of the instrument. So, it bears repeating: to zero a current measurement, you open the circuit.
In reality, the instrument isn't providing a 0.15 uA current offset, it is providing a 12 to 16 uV voltage offset (according to your measurements), which you can mentally account for as a source of error, along with the burden voltage caused by the shunt resistance (which is almost certainly much more than a few uV). If those microvolts of offset really affect a 'real' current measurement (not leads shorted together), then you need to consider an instrument different than a bench DMM. When measuring uA, the burden voltage will typically be much higher than a few uV, so that offset is not the largest source of error.
I suggest reading Keithley's "Low Level Measurements Handbook", now in 7th edition I believe.