If I am understanding you, it sounds like the shotgun is working as it is supposed to (at least in the respect about which you express concern). The sequence described in your fifth paragraph does not have enough information to positively determine whether a fault exists. The trigger could have been pulled before the snap caps were inserted in the magazine, decocking the shotgun.
The action release button is to allow cycling the action to unload unexpended shotshells, or to load the first shell after dry firing. The action is locked closed when the striker is cocked to prevent accidental unloading of an unexpended shell, and for safety. It should not matter whether the shotgun is loaded.
Open and close the action, put one or more snap caps in the magazine, open and close the action (without your finger on the trigger) to load. It should not allow further cycling without the action release being depressed. If it does, take it to a smith or back to the store.
One more thing, "loaded" means "with a round in the chamber." Having cartridges or shells present in a magazine without one in the chamber is not considered loaded, operationally-speaking at least, legally may be another matter depending on where you live.