I don't know how others evaded the draft but I can tell you how good, ole CONNECTICUT Yankees did it.
The draft was instituted in the summer of 1863, shortly after Gettysburg-Vicksburg-NYC draft riots. (Maybe that's why the governor issued pistols to the officials in charge).
So about 12,000 were drafted. 8000 were exempted: unfit, breadwinner, only son, etc. Another 2248 paid poorer guys as a substitute. So far we've got 2248 able bodies. Another 1750 inductees brought the total to 3998 draftees.
Well a bunch of the 1750 draftees were ticked off and burned down the conscript camp in New Haven. These guys just disappeared. POOF !!!
So now the group is headed down South --- to NYC, to Newark, to Philadelphia on board trains. In stages the subs and draftees "jumped ship". A couple of times they assaulted guards.
They left the train stations and most likely headed WEST. To communities in Illinois, in Minnesota, the territories, to California. New name, new identity, new life, new career.
So how many CT draftees honorably served? 248 or about .02.
Reference: CONNECTICUT IN THE AMERICAN CIVIL WAR Slavery, Sacrifice, & Survival, Matthew Warshauer, 2011.