Firstly, well done Unstuck. It takes courage to honestly research your convictions (not just about religion, but about anything) but the personal satisfaction of doing so makes it all worth while.
There are many ways to research what is the correct date for the fall of Jerusalem. I think whether one way is more convincing than another depends on the individual. For example Wifibandit has an outline that shows, using Watchtower's own publications, that it must have been 587 BC.
Stone tablets such as VAT 4956 would be conclusive proof in most people's eyes. VAT 4956 was a record of astronomical observations made (according to the tablet) in the 37th year of the reign of Nebuchadnezzar. The astronomical observations recorded as described only happen once every 20,000 years, and would have occurred in 567 BC. Given the bible says Jerusalem was destroyed in the 19th year of the reign of Nebuchadnezzar, the date of destruction must have been 586 BC (not 607 BC).
Watchtower did a job on VAT 4956 a few years ago. In an article, they argued that the Babylonians didn't have clocks (crap) and the names the Babylonians gave to planets was uncertain, and therefore excluded most of what was written on the tablet. They then fitted the remaining observations into their chosen year; they could have probably fitted into any year, at that point. Worse, from memory, the Online version of the article did not even mention the dodgy assumptions Watchtower made. The effect is that any ultradub will throw that article at you, if you try to raise VAT 4956.
You can also show Watchtower chronology is wrong in other ways, by for example, going through Egyptian records of when Pharoah Necho was on the throne.
Personally, I think the easiest way for someone at home with nothing more than the internet, to prove Watchtower is wrong, is to use Ptolomy's canon.
Ptolomy's canon is really just a list of observed eclipses going back to 747 BC. The list was maintained and handed down through the ages. There were no "BC" dates back then, so the year portion of the date of an eclipse was recorded by reference to the relevant ruler of the major power in the near east, be that Babylonian, Persian, Ptolomeic or Egyptian, and the year of his reign. Because today we can accurately calculate when eclipses would have occurred even thousands of years ago, we can check whether it is accurate, continuous, and assign exact dates for each reported eclipse. Because the purpose was to record eclipses, not act as a calendar, it does not describe when during a year a king took power, and so any date derived from it can be up to a year out. (Short-lived kings that lasted less than a year often don't get mentioned, for that reason.)
As a result, anyone can simply look up Ptolomy's canon and see that Nebuchadnezzar came to power in 604 BC (+/- 1 year) and Cyrus the Great conquered Babylon in 538 BC (+/- 1 year).
I like this approach because once you understand Ptolomy's canon, you realise it is plain unarguable, it is easily checked at home without relying on experts, and as far as I am aware, Watchtower has not yet concocted any rebuttal.