For the linear actuator, a single power supply and a single Double-Pole Double-Throw (DPDT) relay is all you need. This power supply should be always on.
But to control the DPDT relay, which will switch the polarity of the power supply and make the linear actuator go the other direction, you'll need another (much lighter duty) power supply.
So get a DPDT relay, then wire it up so it is effectively just switching the polarity of the linear actuator. You can see the example wiring here (
https://www.firgelliauto.com/products/12-volt-double-pull-double-throw-relay#ptab-technical-drawings).
Then for the DPDT relay, wire the coil leads to an appropriately sized power supply in your EB4, and when you turn the power outlet on, you'll power the coil and flip the DPDT relay. When you shut the power off, the DPDT relay will flip back and the linear actuator will go the other way.
Just poking around Jameco, it looks like you can get an appropriate relay for $6 and a light duty power supply to go along with it for another $5-6 (though I'm sure you'll have a light duty 12v DC adapter lying around). Then you just need your beefier power supply for the linear actuator and you are good to go.