Possibility of using 5A 5-220VDC Solid State Relay for switching 120VAC String Lights
The references you found clearly show that the Mager GJ5FA-L is for use with DC and NOT for AC, so this is NOT suitable for your application.
The fact that it's called a Solid State Relay does NOT imply usage for AC or DC. Even though you can't find a datasheet, if the data pointed to is correct then the output pins show connection for +ve and -ve. 

 

If the output circuit is a high voltage FET then it would have an intrinsic diode which would explain the comment that it conducts all the time if the polarity is revered. 

If you want to switch AC voltages, then you need an SSR like this, though this is rated for only 2A at 100 to 240V AC.(50/60Hz).