Door catch striker help

I can't think of any reason for the observed behavior that you can't move the strike straight out. I would fully loosen the screws or even fully remove them to make sure the backing plate with the threaded holes is free to move whereever it needs to.

Keep in mind that in addition to in & out and up & down, you also have the ability to rotate the strike, so perhaps that is your problem. Also lube the latch mechanism to be sure everything is working smoothly.

Make sure the hinges are adjusted properly. With the strike fully removed, close the door and make sure the body lines line up between the door and the quarter panel, both up & down, and in & out at both the top and bottom of the door. You shouldn't be using the strike adjustment to correct for misaligned or sagging hinges.

To get the strike good and tight, be sure you are using the right size phillips screwdriver. Not sure the number, but you need a big one. You should have conical lock washers on the srews and I would lubricate the screws so you get more clamping force for the same amount of torque.

Hopes this all makes sense.