If you are switching 12V DC, then yes I think it is the wrong relay. I am a little unclear what voltage you are using. You show on your diagram a DC voltage with the + and -, but describe a transformer which is AC. I have assumed (possibly wrongly) that you are actually using a 240V AC to 12V DC power supply.

Do you have the specification for the inputs to the drivers ? AC or DC and how much current. When you know this you can the choose a better relay or other driver.