TLDR: Adams was writing about the Massachusetts State Senate, the upper house of the state legislature.

As well as I can read Adams' handwriting, she describes the rebels as:

... Desperados, without conscience or principles, have led a deluded multitude to follow their standard, under pretence of grievances which have no existence but in their imaginations. Some of them were crying out for a paper currency, some for an equal distribution of property. Some were for [illegible] all debts. Others complaining that the Senate was a useless branch of Government, that the Court of Common Pleas was unnecessary, that the Sitting of the General Court [i.e. the state legislature] in Boston was a grievance.

This's obviously a biased description, but she does correctly list their grievances: Shay's Rebellion was largely over debts owed by farmers in western Massachusetts, a shortage of currency which was perceived as favoring creditors (mostly in Boston), and a state government which was perceived as favoring the Boston-based establishment. All of these grievances were against the Massachusetts state government. (Under the Articles of Confederation, states largely controlled the currency as well.)

What's more, the rebels were largely correct that the state government was however-unintentionally structured to favor Boston. At the time, representatives were paid by their constituents. Impoverished and indebted farmers in western Massachusetts often couldn't pay their representatives (let alone their travel expenses), so they were forced to often go unrepresented. Their complaint about the Senate was probably related to this: if they could hardly pay a representative in one house of the legislature, a second house would simply add an extra burden.