Bill, According to the Plans & Specifications of the L.C. Smith Shotgun, there are only 3 springs in the trigger. The first spring mentioned is the spur lever spring. The second spring is the Weight spring. The third spring is the trigger spring.
I'm wondering if the trigger spring is weak and not returning the trigger properly?
There is a total breakdown, piece by piece, with an explanation of what each piece does and the specifications for that part. The book states that after the recoil and rebound cease, the slightest release of the trigger will permit the spur B to disengage from the hook and rest upon the elevated stud F, thus, bringing the parts into position to fire the 2nd barrel instantly.
I think if you would get a copy of the Plans & Specifications, you would be able to troubleshoot your trigger.