This is true! Most of the regiments I described above were raised for 1 year or less.
Also correct -- another very prominent example is the 26th New York Cavalry, which contained a battalion each from Massachusetts, Vermont, and New York.
And this would especially be true with the units raised for 1 year in February and March, 1865 -- whether "veteran" units or not, many DID contain men with prior service. For instance, almost every field officer in the Ohio state militia had first seen service in the Ohio volunteers.
Which then brings up an interesting question -- can regiments such as the 2nd Maine Cavalry be considered "new"? One of its qualifications was that its recruits needed to be veterans, and as this followed the muster-out of the 21st-28th infantry regiments, multiple field officers came from those Port Hudson veterans.
And indeed they were -- when reading the histories of these units, one will find that few served in the field. Excepting, of course, maybe ten of the Ohio units and five of the New York units, most served in garrison duties from Winchester to Leavenworth.
The outlier to all this, of course, is the 1864 National Guard, which I believe was the first federalization of US state militia on a large scale. In relation to the OP, compare what Lincoln did with the National Guardsmen with what Jefferson Davis did with the Arkansas, Georgia, and Mississippi militia. The National Guardsmen were allowed to muster out peaceably, while the Confederate militiamen were given two choices: transfer to Confederate service or be conscripted into Confederate service.