I find absolutely no record of Seth Shaw Jr and Letitia having a son named John Shaw 2nd in Ware, Palmer, Western (Warren), Hardwick, or Brimfield vital records.
Seth's father (Seth) and mother (Jane/Jean) had a son named John born in Palmer in 1750. But I am finding no record that I'm positive is for their son John beyond his birth. The closest "John Shaw" marriage date I can find to Seth and Jane's son John is a John Shaw married to a Ruth Davis, but they noted as being married 5 years after John Shaw 2nd (son attributed to Seth Jr and Letitia) was (supposed to have been) born.
Note: History of Ware, Massachusetts by Authur Chase - On page 275 a land transaction is noted for John Shaw, 2nd in 1806.
At this point I'm leaving the parental connection intact even though I'm positive it is incorrect as I still have a few historical records of Ware and Palmer to read through.
Note: According to History of the Town of Palmer by Josiah Temple. On page 548 the "John Shaw's" of Monson and Wales are descendants of a John Shaw that settled in Grafton, and moved to Wales in 1752.