The 31-year-old mother of three married cage fighter Reid, 34, in Las Vegas last week.
In her first television appearance since she and Reid tied the knot, Price, also known as Jordan, told GMTV the couple were "so happy" and questioned why people "picked on" them.
Price said of the rumours about the marriage: "If people really believe this there has got to be something wrong. It's a joke."
She claimed she had been keeping her profile "quite low" recently but accused newspapers and magazines of writing "untrue stuff" to sell copies.
"People are treating me like a serial killer, I'm a villain," she said. "What have I actually done wrong?"
The model also denied that she and Reid had agreed a deal for photos of their wedding ceremony in a Las Vegas chapel last Tuesday.
She said the chapel filmed the ceremony, which she described as "not tacky at all", and declared: "As it stands we have not done a magazine deal."
Of the occasion, she said: "It was fantastic. We had a brilliant day. It was just like a dream."
Price and Reid confirmed they plan to celebrate their marriage with a blessing later this year for their families and so her daughter, Princess Tiaamii, can be a bridesmaid.
Asked if she would like to be known by her married name, she replied: "Absolutely Mrs Reid, yep."
"I am in such a happy place," she added. "It's so great. New year, new beginning and a new family life to start. We are so excited."
She also said the couple are trying for a baby.
"Me and Alex so want kids. We are trying, so let's hope, let's hope."
Price added later: "I can't wait to get pregnant."
She praised Reid's relationship with her children, while also singling out her former husband Peter Andre's skills as a father.
Price dated Reid for around seven months before their wedding.
Much of their relationship has been played out in public thanks to a string of TV appearances and extensive newspaper coverage.
The pair briefly split last year, after Price dumped Reid on live TV during her exit interview on I'm A Celebrity... Get Me Out of Here!
But the model said she had been "heartbroken" at breaking off the relationship, and the couple were reunited.
During today's interview, Price responded to a series of questions from presenters Ben Shephard and Kate Garraway about the speed at which she married Reid and her apparent courting of publicity through television documentaries and autobiographies.
Price thanked the pair, who littered their interview with laughter and light-hearted banter, for helping her "clear up the horrible rumours".
At one point, Garraway said: "You are brilliant at keeping everybody going. I don't know whether it's design or whether it's luck that this happens to you.
"As soon as you and Pete split up, people said 'Oh, God, what's going to happen next?' Then suddenly there you are on I'm A Celebrity... there you are dumping Alex live on air, now you are married to him. It's just like, wow."
Price replied: "I know the magazines and papers don't show it but I have actually been keeping quite low."
She said she no longer buys papers but is told about stories, adding: "They are still writing untrue stuff. I just don't know where they get it from. It's a joke."
Shephard told viewers that Price arrived on set with a camera crew, prompting him to ask if there will be another fly-on-the-wall series - which Price said will be screened on Thursday.
Garraway later asked: "The speed of getting married, how could you have known he was right? How could you just suddenly know?"
Price recounted how Reid had proposed last September and she had said yes, but kept it quiet, before Shephard cut in to point out: "Then you dumped him?"
"I haven't had the time to explain it to but you will see in the show," she answered.
"We were both poisoned in our minds about things, people around us, and anyway we are back together."
Price lashed out over an interview with Andre last week in which questions were put to him about his children.
During the interview with Sky News, Andre was shown archive footage of Dwight Yorke, the father of Price's son Harvey, criticising Andre for wanting to adopt the child.
Andre, who is the father of Price's two other children, was asked how he might feel if Reid said he wanted to adopt the children.
Pointing a finger, Andre hit back: "No-one is going to take my kids away from me - nobody."
Price told GMTV Andre was a "fantastic dad" and added: "I thought it was so cruel, so out of order.
"They can't compare Dwight and Pete at all."
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