When LED Xmas lights first made an appearance on the scene about 3 years ago, I had problems with them not shutting off completely. Even though they were supposed to be off, they still glowed faintly due to leakby current. So I took a unused 120v to low voltage (i.e. 6, 12 volt) transformer and cut the low voltage side to about 2 inches long and taped off the ends. I then plugged it into the same power strip with the lights to give it a slightly larger load than the Xmas lights. This solved my problem of them not shutting off.
Not saying that this will help your problem but it is worth the try. Just plug in a low voltage transformer without cutting off the secondary wire. If it works, then cut off the load wires and tape them.
I am sure that you have an unused transformer from an old video game or phone.