LM350 voltage rises with capacitor
The purpose of your C2 is to create a slower turn-on and even slower droop as a result of overshoot since there is no pull down driver, just pullup. (Darlington NPN)  

A secondary misleading advantage is to reduce massive full wave rectified in input ripple to output by some minor amount.  The disadvantage is it seems to raise your Vmin adjustment range.  This the point when R1 is near max but maybe not exactly 0 Ohms and the output rises for you. (corrected)

My suggestion is to move C2 to output
You do not report an input or output ripple issue.
Then move diode to in-output to protect against input short circuit damaging output.  (opt.)

Also, make sure you have adequate heatsink+fan if you plan to use 3A at 10V with an 8V drop or 24 Watt dissipation.



SPEC says MINIMUM LOAD CURRENT = 10mA max  ( not 10V over 10k=1mA)

Low ESR is not a problem for C3 as this is a bipolar LDO not a MOSFET LDO which can suffer from low loop gain from excessive low ESR.