My husband's g grandfather, c1857 - 1938, took the name of another living man to avoid the draft. He married in Cekiske in 1880 as Shlomo Levin b Movsha from Sirvintos. Shlomo's father had died in Sirvintos in 1867 at the age of 50 of typhus. Presumably, Shlomo or his family needed the money, so he sold his identity. Below is the list of children born to Shlomo Levin, and at the bottom is my husband's grandmother born to the man who was really Yeheskel Gordon, son of Mordecai and Chaya Riva. We inherited a tree with those names and my husband, his cousins, and his father all had the middle name Gordon to keep the surname alive.
Shlomo Levin used the name Samuel Levin when he moved to New York. We don't know if he was really Yecheskel Shmuel, or if he assumed the name of his grandfather who had died in 1868. However, he didn't want to be buried under the name of Samuel Levin. While the surname Levin is on the headstone instead of his birth surname Gordon, his given name was changed to Charles S., and his Hebrew name is Yechetskl son of Mordecai.
We were never able to find Mordecai and Chaya Rive Gordon and son Yecheskel in any records, in part, because we didn't know where the family had lived. It wasn't until my husband had a very strong DNA match to two Gordon siblings whose g grandfather had emigrated from Odessa that I was eventually able to find their death records in Odessa. Both death records said Mordecai and Chaya Riva were registered in Moletai. Knowing that, I am pretty certain that I have found Mordecai and Chaya in two revision lists, but no sons were listed, only a daughter Ester who was three years old in both 1851 and 1858. I suspect that in both lists the daughter was a son dressed as girl. I don't know how they hid their sons when they were too old to pass as a girl. The first Ester was probably Solomon born c.1849 who emigrated from Odessa, and the second Esther was either another brother, possibly a Yusel I saw in Odessa records, or Yecheskel supposedly born 1857. Unfortunately, Solomon, his mother Chaya Riva, and possible brother Yusel are not in the extant 1897 Odessa Census.
So yes, stories about men avoiding the draft via false papers are true, and in the case above, for three or more years both men were living in Lithuania, but not the same area.