Sounds like there might be something on the safety that's is loose, interfering with the trigger. when you have it apart for cleaning, see if the safety is solid or a bit sloppy, seems like it may shift/slide in front of trigger, I've read some posts on other forums, seems like it's a common thing with SX4. I'm not an expert, but your details make me want to check that safety.
Here's a "Hail Mary" attempt as assistance.
Be sure that the hammer is cocked and the safety in the "ON" position when inserting the trigger assembly into the receiver. Then, once the trigger assembly is fully assembled into the receiver with all pins in place, push the bolt release button and see if the bolt slams shut on the empty chamber as it should.
Then, push the safety to the OFF position and then pull the trigger to drop the hammer on an empty chamber. Then pull the bolt handle fully to the rear to lock the bolt in the rearward position. If everything works as it should up to this point, then I think you've got it assembled correctly.
If my suggestion above doesn't solve the problem, then the problem is probably in the safety switch itself. I believe that gun has a trigger block safety located in the trigger assembly.
There might be metal shavings or other obstructions inside the trigger assembly of the new gun. In that case, flood the safety switch with almost any kind of liquid cleaner or lubricant (mineral spirits, kerosene, gun oil, or whatever) in an attempt to flush out and/or lubricate the safety. Work the safety back and forth repeatedly while it's well lubricated. That may flush out the safety switch and free up the operation of the trigger.
If all that fails , Gunsmith is an option, the more i read up on these shotguns, Some users have had that gun fire by itself , with no birds in sight.
get it checked!