I have a Jesse Shaw of Wilson Co. Tennessee, probably older than your Jesse. I estimate his birth about 1780 in North Carolina. He died in Saline Co., Illinois in 1837. I have a copy of his will.
His will list just one son---Leroy Wesley Shaw, and his wife Sally and his brother-in-law John Cain. However at his death he had 3 other sons under the age of 10, and a pregnant wife. Leroy was born in Wilson Co. Tennessee in 1819. I believe his mother was Mary Pack who was married to Jesse in Wilson Co. in 1819.
One of Jesse's grandsons had a bio written about him that lists Jesse's name as Amos. It also states that Jesse came to Illinois to what is now Saline Co., Illinois in 1830. There is one Shaw in the 1830 Saline Co, census---the right age, but named Gabriel Shaw.
There were 3 Jesse's in the 1830 Tennessee census. Henderson Tn. age 20-29 too young,
Roane Tn age 50-59 right age and
Wlliamson Tn age 60-69 too old.
In Wilson Co. Tennessee there was a Jeremiah Shaw and a William Shaw listed in 1830.
I sure would like to know more about my Jesse Shaw. Anything you could share would be welcome.