The two politicians both promised to hold the UK government’s feet to the fire on Brexit if they are elected to the helm of the party.
But while Mr Nuttall struck a conciliatory tone and pitched himself as the “unity” candidate, Ms Evans used her bid to launch a scathing attack on the current leadership.
She said Ukip needs to shed its “toxic” image and warned it risks becoming a Donald Trump-style party under the leadership of Nigel Farage and his allies. She accused her leadership rival, Raheem Kassam, of being a “far-right” candidate that will take Ukip in the wrong direction.
Ms Evans said: “Our future as a political party in Britain does not lie in that far-right wing. I don’t see a groundswell of opinion in this country for more far-right wing policies.
“I don’t see a groundswell of opinion for the right to bear arms in America.”
Ms Evans said she “absolutely” thought Mr Kassam – who has won the backing of party donor Arron Banks – will take Ukip in a far-right direction but “our members don’t want that”.
She said: “We’ve taken a lot of stick in Ukip because perhaps we have had a slightly more toxic image than we should have had. And our members, the ones that are doing the campaigning, have felt the brunt of that – being abused, being physically and verbally assaulted on the streets.
“They don’t want to have a fresh injection of toxicity that’s going to make it even more difficult for them, they want policies that help us win.”
Mr Farage is close to Mr Trump, appearing at a rally in the US for the controversial presidential candidate.
Ms Evans, who wrote her party’s 2015 manifesto but has since fallen out with Mr Farage, was unable to stand in the last leadership election after being suspended from the party for disloyalty.
But she tried to brush off claims that her unpopularity with the Ukip leader and his close ally Mr Banks would harm her chances of being elected to lead the party.
Mr Farage hit out at Ms Evans’ remarks, telling ITV’s Peston on Sunday: “For her to talk about the party being toxic, for her to already declare one of the candidates who is running, Raheem Kassam, as being far-right, I don’t view this as being a very good start.”
Steven Woolfe, the front-runner to replace her, also left the party in the wake of an altercation with another party member after which he ended up in hospital.
Mr Nuttall, Ukip North West MEP, said the party faced an “existential crisis” and could disappear unless it unified.