Ruling Conservatives and the Labour opposition supported a call for chief executive Barry Keel to examine why an initial letter from the Post Office in July 2007 was not answered.
He will also investigate a letter sent by the Post Office in January, which offered south-west councils a confidential meeting on the closure programme, but received no reply until June.
Opposition leader Tudor Evans (Lab) said a scrutiny group had twice been told that no correspondence had been received.
A Royal Mail spokesman confirmed the lengthy gaps in the council’s responses to its external relations manager Arthur Magee.
Cllr Evans said: “Either Mr Magee did not send the letters he says he did, someone has cocked up, or someone has misled the scrutiny group.”