How to deal with a DC adapter with irregular voltage output?
What is the voltage and current rating of the supply? Is there any other load on the supply? Is the supply a regulated or unregulated supply?

Most of these supplies require a minimum load before the output voltage is within
a specified range. Other than the Schmitt trigger, was there any other load on the supply? If not, you are probably not loading the supply to the specified minimum current.

In most applications these supplies are followed with a linear regulator to provide a stable voltage. You still need a minimum load and some filtering capacitors. 4000&nbsp;µF is excessive and possibly dangerous. Typical capacitors are in the 1-100&nbsp;µF range. 

The size and type of capacitor is dependent on the linear regulator. Follow the manufacturers recommendation exactly. Some devices have very specific capacitance and effective series resistance (ESR) requirements.