I just installed a similar item three weeks ago.
I installed it and had the same thing happen. I unplugged it and plugged it into the other wire and that solved the problem.
FWIW, I hope you didn't cut the OEM wires? You can use a male and female blade connector on the flasher unit and make a short extension wire for the other side also using a male and female blade connector.
In addition, is the unit you purchased for use with LED bulbs only?