It seems relevant here that the late 1770s and 1780s was also the same period when the United States was governed under the Articles of Confederation, rather than the Constitution we have today (see https://en.wikipedia.org/wiki/Confederation_Period).
The Articles of Confederation established a much weaker central government than the later Constitution would. So it would make sense, in a regime where the several states were still regarded as near-sovereign entities, that people would look to the founding documents of their state as the guarantors of their liberty. And it also makes sense that, when the weak central government proved unable to meet the challenges of the time and people started thinking about replacing it with a stronger one, new attention would be drawn to documents like the Declaration that spoke from a national perspective.