Yes, I have an example of adoption to avoid the Tsar’s army, which I've told in a family story on the Byerazino/Pahost KehilaLink. The details are reported here:
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The original family name was Feldb’in, but the eldest son, Herschel, used the surname Rabinowitz, as did his descendants. I first thought this stuff about avoiding the draft was just a ‘fairy tale’ but in fact it truly happened. Herschel’s brother-in-law Louis Katz, who was well connected, arranged for his adoption by a man named Rabinowitz who had female children but no sons.
I verified the original family name thanks to a hand-written history, left by a member of the immigrant generation. If I had not traced all possible living descendants of the original family, and had I not been fortunate enough to meet the daughter of the early ‘family historian' I would never have known the family’s true surname, since Herschel continued to use Rabinowitz, even after immigrating to the US.
So regarding your question about how to proceed, based on my experience, I suggest you start by finding all living descendants and asking for family memories.
I can also strongly recommend taking one or more of JewishGen’s guided, online research classes. I received invaluable personal assistance I from Nancy Holden and Phyllis Kramer (Z”L) - without them, I’d have gotten nowhere!
Once I discovered the original surname, revision lists (original Russian census records) available through the Belarus SIG and the Igumen District Research project allowed me to trace the Feldb'ins and their ancestors back to the late 1700s. Unfortunately, I still have one more brick wall to overcome: no matter how hard I’ve looked, I have never been able to identify the Rabinowitz family that saved Herschel from the horrors of the anti-semitic military.
Carola Murray-Seegert, Ph. D.
Coordinator, Igumen District Research Group
Manager, Byerazino/Pahost KehilaLInk
Manager, Moskva KehilaLink
Researching: FELDB’IN, FELDBAUM, KATZ, RABINOWITZ in Pahost and Byerazino: LIFSHITZ, SHEFTEL in Shklov and Moskva; COIN, FELSER, SCOLL in Tauroggen, Baltimore, Chicago and San Antonio.