The finance secretary has lost support from several SNP politicians after saying she would have voted against gay marriage had she been an MSP in 2014.
Ms Forbes has now taken to social media in a bid to reset her campaign.
It comes after she hit back at John Swinney, who questioned if her views would be appropriate for an SNP leader.
Ms Forbes, who is a member of the Free Church of Scotland, also said she believed that having a child outside of marriage was "wrong" according to her religious beliefs when she took part in a series of interviews on Monday and Tuesday.
There had been speculation over whether Ms Forbes would continue in the race after a series of MSPs withdrew their backing but she has made clear she intends to fight on.
She wrote in a statement on Facebook and Twitter on Thursday: "I feel greatly burdened that some of my responses to questions in the media have caused hurt, which was never my intention as I sought to answer questions clearly.
"I will defend to the hilt the right of everybody in Scotland, particularly minorities, to live and to live without fear or harassment in a pluralistic and tolerant society.
"I will uphold the laws that have been won, as a servant of democracy, and seek to enhance the rights of everybody to live in a way which enables them to flourish.
"I firmly believe in the inherent dignity of each human being - that underpins all ethical and political decisions I make."
Mr Yousaf was targeted by opposition MSPs at First Minister's Questions, with Scottish Conservative leader Douglas Ross claiming he had been "useless" as health secretary and had "made the crisis in Scotland's health service much worse".
Scottish Labour leader Anas Sarwar also took aim at the health secretary, saying he had "failed" and his recent NHS recovery plan was "more about spin than substance".
Mr Sarwar asked: "Does the first minister really believe that the man responsible for failing Scotland's NHS should be responsible for our country?"
Ms Sturgeon stressed again the decision on who the new leader would be was "one for my party".
But she sought to defend the health secretary's record, saying: "Since Humza Yousaf became health secretary - and this is what Anas Sarwar fails to mention - there have been, I think, three further waves of a global pandemic that have affected health services all across the UK, Europe and the world, that's not something that can just be ignored."
Following the session at Holyrood, Mr Yousaf said: "It's quite telling that my opponents want to attack me, want to discredit me quite personally and not focus on any of the other candidates. I think that tells you everything you need to know."
Mr Yousaf also praised Ms Forbes and did not rule out giving her cabinet position if he wins the contest despite seeking to set himself apart from her views on issues such as gay marriage and gender reform.
He said: "I think Kate is extremely talented, extremely able, I think anybody would want to see her at the heart of Scottish politics. I think she's got a lot to give to Scottish public life, including in government."
Ms Regan, who will formally launch her campaign on Friday, has called for an end to "mudslinging" in the leadership contest.