Nabu-kudurri-usur (Nebuchadnezzar II), was the oldest son of Nabu-apla-usur (Nabopolassar), founder of the 11th dynasty of Babylon. From the records, we know that he was born c634BC and died c562BC (aged 71/72). He succeeded his father to become king c605BC. (the "c" in all these dates stands for "circa" or "about")
You are right that we use a number of astronomical events to provide fixed points that we can use to tie the various near eastern chronologies together. A well known one is the "Assyrian eclipse" of 15 June 763BC (although even that date is not without controversy!). As I understand it, and as you say in your question, the "saros" texts record another solar eclipse in the 32nd year of Nebuchadnzzar's reign. Now, I think that has been suggested to be an eclipse in either 573/572BC, which would correspond to Nebuchadnezzar II becoming King of Babylon c 605BC.
Now, you're quite right that, as with many other ancient calendar systems, the Babylonians faced a challenge of making their calendar of twelve lunar months fit with the solar year of about 365.25 days. This has added a further complication to matching events recorded in Babylonian records to exact dates in our modern calendar. A bigger complication is that most events aren't actually recorded with a date, just a regnal year - e.g.
"In the 32nd year of the reign of King Nabu-kudurri-usur ..."
This is why dates are usually given as circa ...; When I see a text that says something like
"Nebuchadnezzar II became King c605BC",
I read it as
"Nebuchadnezzar II became King in 605BC, give-or-take a year or so".
So, yes, Nebuchadnezzar II could well have become King in a different year from 605BC, but 605BC is our best guess; and writing the date as c605BC makes it clear that we are not certain that the event actually happened in 605BC.