I have the 0.1uF capacitor attached across the output, but the datasheet's example shows a 10uF capacitor across both the input and output of the regulator. Is the issue here with my capacitor not being big enough? I'm stumped as to what is going wrong here.
The datasheet for the LM1086 says "The recommended load/decoupling capacitance is a 10uF tantalum or a 50uF aluminum. These values will assure stability for the majority of applications"
It is well known that inadequate load capacitors can cause instability and failed regulation, so therefore please add a 47uf electrolyte (a 100uf electrolyte capacitor would be very safe) and your problems should disappear.