I've not heard of this before, but there might be some inspiration there somehow. Early modern Poland-Lithuania was an elective, limited monarchy. But I'm not sure how much Polish influence there was in the Thirteen Colonies at the time. More obvious inspirations for American democracy were:

-the Dutch Republic, via the colony of New Netherlands

-Quakerism, via the colony of Pennsylvania

-the Iroquois Confederacy

and finally the British political system itself, which was already a limited monarchy with a mostly parliamentary government as well. Compared to most continental European monarchies Poland was indeed relatively democratic, but compared to Britain it wasn't that special. And if the founding fathers were to look further afield, why not be inspired by Switzerland?

So in short: there are too many obvious examples of democracy or constitutional government that the Americans could draw on to place much emphasis on Poland.

No. There were some Polish volunteers who fought in the Revolutionary War, but there's no connection to Polish governmental theories.

If we look at the Declaration of Independence, the Articles of Confederation, and the Constitution, the clearest influences on them are English writers in the Whig tradition.

In the lead up to the outbreak of war, and during the American Revolution itself it was the same--American thought and theory aligned most closely with English Whig thought and theory, and in fact Americans who resisted government policies in the time after the French & Indian War primarily called themselves Whigs and their opponents Tories after the two divisions in England.

The most well-known of the Whig writers today is John Locke, but others were just as influential in the 18th century. These include John Milton, Algernon Sidney, Thomas Gordon, Francis Hutcheson, Catharine Macaulay and many others.

The root of Whig thought was that it was lawful for the people to resist laws if all other efforts to correct them had been tried, as long as that resistance did not constitute an attack on the crown itself. This is one reason why even as late as the spring of 1776 the Continental Congress continued to refer itself as loyal subjects of King George III and did it's best to direct their arguments and attacks against the specific laws, or against Parliament or "wicked ministers".

To quote from Pauline Maier's "From Resistance to Revolution"

A readiness on the part of the people to resist, they [Whig writers] argued, discouraged usurpations and made disorder less likely. Seditions and rebellions were more frequently caused, Hutcheson wrote, by "the opposite doctrines giving unbounded licence to vicious rulers." If princes knew their abuses would encounter resistance, said Thomas Gordon, they would never encroach; and for Hoadly, too, a universal accord on the right of ressistance "would entirely prevent the very beginnings of evil." It was the doctrine of nonresistance in the Turks' religion that had caused their enslavement, according to Cato's Letters; and Robert Molesworth in his *Account of Denmark (1694) warned that "the Doctrine of a blind Obedience, in what Religion soever it be found . . . is the Destruction of the Liberty . . . of any Nation." In short, passivity on the part of subjects irresistibly tempted princes to seize arbitrary power, and this just as invariably induced disorder. "All disputes about right," Sidney wrote, "naturally end in force when justice is denied." There was a further effective curb on license implicit in any defense of the right of resistance, Hoadly claimed, for "he that tells [men] the Liberty they make take, warns them against taking more.' Finally, spokesmen for resistance could not be responsible if men misapplied their principles. Evil men "disposed to public Disturbances" would find excuses in any case.

Here's a bit about the various men quoted by Maier in this paragraph:

Francis Hutcheson was a Presbyterian preacher who lived from 1694 to 1746 and was an important figure in the English Enlightment thought.

Thomas Gordon was a writer and historian who lived from 1691 to 1750 and published a paper called The Independent Whig

Robert Molesworth was a British Lord who lived from 1656 to 1725. He was a strong supporter of William of Orange against James II.

Algernon Sidney lived from 1623 to 1683. He was a member of Parliament who wrote an influential political treatise titled Discourses Concerning Government. Unfortunately Sidney was accused of plotting against the king (Discourses was used as part of the evidence) and executed.

Bishop Benjamin Hoadly lived from 1676 to 1761 and was prominent in Whig political thought, publishing many papers and letters.

The last quote about evil men is from John Locke, probably the most famous of the Whig thinkers.