Though firing a round is dependent on the trigger releasing the striker, ejecting and chambering a new round generally happen with the trigger disconnected.
I suspect you have a problem with one (or more) of the following:
Trigger bar spring
Trigger bar
slide
cocking cam
Some of these are best checked by removing the side plate of the frame, but before doing that try checking the following:
Remove the slide. Look to see if the trigger bar spring is properly seated in the frame and that the end of it is under the trigger bar.
The spring in the following photos is correctly seated and positioned. The trigger bar in the photos is correctly positioned by the trigger bar spring. In this condition, pulling the trigger bar should move the cocking cam.
If the cocking cam moves, let's look at the disconnector.
Take a small tool (screwdriver, butterknife, etc.) and press the disconnector down until the top of it is below the level of the frame, then release it. It should immediately return to it's full height.
Now pull and hold the trigger. The disconnector should have moved foward from it's previous position. Now, still holding the trigger to the rear, push the disconnector below the edge of the frame. This should allow the cocking cam to reset (you see and hear this happen) and the disconnector should remain down when you pull your tool away. Now, release the trigger; the disconnector should pop back up in it's original position (see photo). There should be an audible click when this happens. Pulling the trigger should cause the cocking cam to move.
If everything is good to this point, it might be best to remove the slide plate. Let me what you find while doing the above and we will continue.
Regards,
Greg