Weird op-amp voltage increase when using AC-DC power supply
The spikes in output voltage may be due to your AC/DC switching supply not being able to regulate its output completely. See: Does a switching supply always need load. Did you check the supply rail with an oscilloscope?

You can solve the poor regulation by loading the power supply more using resistors between the supply and ground. Be aware of your resistors' power ratings and that they will get warm. You should use 1/4W resistors with no lower resistance than 125 ohms (from \$0.25 W = (5.3V)^2 / R\$).