I had a similar problem with my 12 ga 870 when I first purchased it years ago. It would do something similar - when the slide was cycled, it would release an extra round (sometimes more) from the magazine.
At that time I purchased the AGI video on 870 gunsmithing (I've since sold it here on EE) and although it didn't solve the problem, it helped me solve the problem.
The 870 has two spring shell latches (as per your Remington Manual exploded parts diagram), one to the left of the magazine tube and one to the right. One shell latch (I forget which one) stops the next round to be loaded from coming completely out of the gun. The other shell latch stops the round in front of that round from coming out of the magazine.
What I figured out, and did, is I just squeezed the two shell latches inward a little to give them a bit more 'action' or 'bite'. This ensured the next round was caught properly by the shell latch and the problem went away. I've never had a double feed since.
I'm not a gunsmith but this is what I did. My advice is worth the ink it is printed with.
Cheers and good luck.
Fusilier