Hello, thanks for the feedback. It's much appreciated.
I will try to explain how I solved this paradox issue.
You see, to try and fix the watch duplication, there are multiple hints throughout that attempt to explain what's going on.
When a person time travels, the pocket watch is noted to disappear from containment and re-appear with the person in question. E.g., in a few of the logs, the person is found with the watch at a specific time, and the watch is noted to disappear from containment corresponding to when the person arrived in the past. When somebody travels back, the watch goes from where it was to the person's hand. There is only ever a single watch.
The story is also formatted to suggest that the people - are - found before. The system of time that I have suggested follows the sort of 'branching' system seen in films like Back to The Future. Traveling in time will affect the timeline, changing it irreversibly, whereby you simply appear in the past instead of using the watch in the future. If this is confusing, just note how only on the 'long hold' experiment did anyone actually use the watch and not see the result in the past. This was because if you go farther back than your own birth, then, heh… well, I'll just leave that alone. In every other case, the person appears before the experiment is conducted, causing it to not be conducted. But, in this system of time, time doesn't loop back on itself, meaning that no paradox is caused.
I now also see that the source of the disintegration should be made more clear. I tried to hint at it, but apparently, I didn't nudge strongly enough. I'll keep it a mystery in the thread itself, for now, at least, but I'll be making some major edits quite soon.