If it's any help, I know my Lewis family were boat builders in Pittsburgh in 1800 and we are connected to Cardinal John McCloskey of Albany NY by an obituary.
If I can read John Fitch's notes correctly, Alexander Lewis owned the boat patent on the Lewis and Clark boat issued by Thomas Jefferson in 1789.
Then Bishop Talleyrand was the main contact for Judge Robert R Livingston in the purchase of the western lands that was going to be explored by Lewis and Clark.
For sure the Talleyrand family had a close connection to the McCloskey family via Alexander Hamilton and his aid Meriwether Lewis.
So yes, I believe the Randolph family of Thomas Jefferson might be related to Randolph Churchill. I know for sure Randolph Churchill's grand daughter, actress Sarah Churchill, stayed with the Jerome family in Rochester NY.Some people believe it was just a few blocks from the George Eastman House.