I have been researching the Shropshire Berringtons for a friend of mine and may have some info for you

I have found James Berrington christened at Hope Bagot (just over 3 miles from Coreley) on 28th June 1789. This is probably the James you are referring to who had a family at Coreley, though I note you have a slightly different date.

James was the son of William Berrington and Mary Anthony. William was christened at Burford 22 June 1755. Mary came from Richard's Castle. They married at Burford 7 March 1779.
I believe Mary was buried in Ludlow 24 April 1809
Their children :
John - baptised at Greete 20 Dec 1779 (buried 8 days later)
William - baptised at Greete 5 April 1781
John - baptised at Greete 20 April 1783
Thomas - baptised at Hope Bagot 12 March 1786
James - baptised at Hope Bagot 28 June 1789
Mary - baptised at Hope Bagot 3 March 1793
Benjamin - baptised at Hope Bagot 13 Dec.1795

James's father, William was the son of Charles Berrington and Anne Granger at Burford 22 July 1750. Charles was buried at Greete 29 June 1785 and Anne was buried at Greete 12 Jan. 1797

I believe my friend to be descended from the other son of Charles and Anne, John who was baptised at Greete 8 Oct. 1752.

I have assumed that the the Hope Bagot and Greete baptisms are the same family. There's an elememt of if and maybe about it, but I believe that is your family.