Am I obligated to pay back pay for homewners dues when I was initially asked to pay a lesser amount?

It would be necessary to review your association's governing documents to fully respond to this question. Describing the initial assessment as "estimated" might indicate that the actual assessment amount was subject to change later. If the board's reasoning for asking you to pay more is based on an accurate reading of the governing documents, then it would have a persuasive case in a collection action against you.