The duo were part of Adam Hinshelwood's squad last term and will meet their former side next season after the Hornets were promoted to the Bostik League Premier Division for the upcoming campaign.
Newton has enjoyed several spells at Worthing during his career, ending last term on loan at Woodside Road from National League South side Welling United.
After leaving the Wings, the winger was sold on a switch to Horsham following a chat with manager Dominic Di Paola.
Newton said: "I think it’s just the new facilities essentially.
"It’s a new team coming up and there are a lot of prospects.
“I had a chat with Dom and he mentioned a lot of good things about where the club wants to go, so I’m willing to try out with him and go forward with the club.”
Newton will be joined by another former Worthing team-mate at Horsham's new Hop Oast home next season.
Rance was released by Hinshelwood in January before joining Burgess Hill, who were relegated from the Bostik League Premier Division.
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