I think you need to identify what the direction of 3 ports on the oil filter adapter. Most likely it is 2 "ins" & 1 "out". Take the filter off and look to see how the ports connect in relation to the filter. An "in" will put oil into the filter through the many holes on the outer ring. An "out" will have the port connected to the center stem of the filter.
If you filter adapter is 2 ins (sides) and an out (center) like I suspect, then the way you have it plumbed now is kinda incorrect. Your oil would travel through the filter and cooler, but would return to the motor at the rear drain pan port, not the rear of the motor. The port (centered) on the back of the motor is the true oil inlet as this is the center stem location of the stock oil filter. Your current line from the back of the motor ("in" for motor) to the front port (suspected "in") wouldn't be doing anything.
If your filter adapter is 2 outs (side port & middle port) and an in (connected to front drain pan port), then your plumbing is correct. The oil would go from motor through the filter and one line back to the motor (back connection) which most oil will follow (least rsistantance) and the other line to the oil cooler and back into the motor (rear drain pan).
Once you identify the direction of ports on your filter adapter, I think you can eliminate a line and plumb them like this:
Oil comes out of the engine at the front oil pan connection (oil pump location). Keep your line from there to the oil filter adapter. Confirm it is connected to an "in" on the filter adapter.
The oil then will pass through the filter and exit out the center stem. Connect a line from an"out" of the filter adapter (confirm middle port is an out) and run it to the bottom of your oil cooler like you have.
The oil will pass through the oil cooler and exit the top. Connect the line at the top of the oil cooler to the back of the motor (not the rear oil pan connection). The port (centered) on the back of the motor is the true oil inlet as this is the center stem location of the stock oil filter.
This would complete a proper loop. Oil comes out of the motor (front drain pan port), goes through the oil filter, goes through the oil cooler, and then back into the motor (back of motor).
You could then plug the 3rd port on the filter adapter and rear drain pan port.