The six handwritten letters sold for £15,100 (NZ$26,700) at auction, exceeding the £2,300-£3,600 estimate.
One lot, which included the letter describing Prince William's adoration for younger brother Harry, was sold for £3,200 - around five times more than its estimate of £400-£600.
The lot containing the letter about Prince Harry's mischievous streak sold for £2,400, more than double the £600-£900 estimate.
In a letter following Prince Harry's birth, Diana wrote warmly: "This is just to say a big thank you, Cyril, for thinking of us at this particularly happy time, with lots of love from, Diana."
The fondness the princess had for Mr Dickman - who served as a palace steward for more than 50 years - was clear when she penned him a letter following the death of her grandmother, the Baroness Fermoy, in July 1993.
She wrote: "Thank you so much for your lovely letter concerning my Grandmother. Her death came as a great shock, but it has meant an enormous amount to receive messages of support such as yours.
"I often think of you and Mrs Dickman and hope that life is treating you kindly."
In 1993, Mr Dickman was advisor to Sir Anthony Hopkins for his role as a butler in the film The Remains of the Day.
Diana mentioned his star turn in one of her letters to the palace steward, saying, "I wait with interest to see the film".
Princess Diana also wished him a speedy recovery upon hearing he was ill in 1985, writing: "I just wanted to write a quick note to say how very sorry I am to hear you are not feeling well. We missed you yesterday at Buckingham Palace."
The letter remarking on Prince Harry's behaviour was sent when he was a pupil at private Ludgrove School in Berkshire in 1993 and shows he had a rebellious streak at a young age.
Mr Dickman started his career at Buckingham Palace watching out for fires on the roof during the Blitz. He retired at the age of 65 and was made a Freeman of the City of London.
When he fell ill towards the end of his life, Prince Charles visited him at his home. He died aged 85 in 2012 and the letters were uncovered by his grandson who put them up for auction.
The six notes were part of a collection described as "the private letters between a trusted butler and the royal family", which also included photos and Christmas cards.
John Foster, of Cheffins auctioneers in Cambridge, said interest in the collection had been "phenomenal", with inquiries coming from as far afield as Japan, Australia and Canada.
He told the BBC: "One of the nice things about this collection is that it reveals the personal side. It's not just thanking someone for an event, it's things like the handwriting and the squiggly drawings.
"There is also mention of when Diana's father passed away and how sad it was, talking of the children when they were at school, how much trouble they got into, just those personal touches which you never really get to hear about."
