Hi JenniferG,
You are right the difference shouldn't be this much in measurements within the different ranges of the same instrument but in your case it appears so. That is why bdunham7 has suggested to put both of your meters in series while measuring the current. When you'll change range on 61E from mA to uA, the other meter will verify if your 61E has any problem/fault or not. Also, please keep in mind that this meter is geared towards hobbyists not professionals so less attention has given in its design to keep things agreeable within the instrument.
Sadly, manufacturers has stopped including detailed operating instructions in the manuals. I guess it makes their instruments appear less attractive.
I am attaching here three pages from Fluke 8020A multimeter's user manual (
https://dam-assets.fluke.com/s3fs-public/8020A___imeng0000.pdf) that'll hopefully explain you why you are getting such a big difference between mA and uA current ranges within the same instrument. Also, attached here is the schematic of UT61E. You can see that this meter uses 1k for the lowest range in uA position and 10 ohms in mA range. Please do watch Joeqsmith's video in reply #12 regarding burden voltage of this meter.
If you want lowest error in current measurements you can get a meter that has lowest burden voltage. The cheapest meter I know that has lowest value shunts thus have lowest burden voltage is UT61D. It is using 50 ohms (uA range), *0.5 ohms (mA range) and 5 milliohms (A range) resistors as shunts.
Edit: *0.5 ohms current shunt for mA range.