If you don't have a ground reference in the whole circuit (as is likely in a battery powered device) it's good practice to hook it to ground in one end. I'm even doing so to improve the noise in resistance measurements (having a 121GW you are likely to have seen my and other people's posts about that)
Looks like the µCurrent is fine, the offset you are seeing is just noise, which with an ungrounded, unshielded, high gain amplifier (as the µCurrent) you are likely to get.
Run the same measurement as the problematic one but do two things, first ground some point of your measurment (µCurrent low seems fine, both lows are connected together IIRC) and second, start (as you always should) with the highest current range and go up by one. That way, if your measurement range is too low you jump up a range.
While precision opamp inputs are pretty sensitive to abuse and even if they don't release the magic smoke, they will go out of specs. That usually happens when the BE junction is polarized backwards over the zener voltage of the junction, usually at about 6V, and higher currents to damage the device. 4.5V with some limiting resistance doesn't look so bad.
I don't know what you are trying to measure, but if you have a 121GW you could just measure with it (unless you are going to the scope or some other instrumet) which already has low burden voltage, you can monitor it at the same time AND has better input protection, which is non existent in the µC, being one of the weak points of it. Notice that the fuses inside the 121GW would cost as much as the fuses in on offs and around the fuses to actually protect the whole thing you need to include include beefy, low leakage diodes, which is likely why dave decided not to go with them.
JS