Gary Allen, 47, is on trial at Sheffield Crown Court for the murder of sex workers Samantha Class, 29, in Hull in October 1997 and Alena Grlakova, 38, in Rotherham on Boxing Day 2018.
Allen, of no fixed address, is accused of stamping on Ms Class, before strangling her with a ligature, running her over with a car and then dumping her body in the River Humber on October 26, 1997. Her body was found by schoolchildren on the foreshore of the Humber estuary.
He admitted having sex with Ms Class after he had been out drinking with a friend in Hull, but said she had got out of his car angry after a condom they had used split.
Sheffield Crown Court heard on Wednesday how Allen came to the attention of police after he was caught drink driving in the Hessle Road area of Hull in the early hours of July 6, 1998.
The police officer noted how Allen had accelerated when he first saw the officer's marked car, but pulled over when instructed to.
In a statement read to the court, the officer said: "I asked him why he had accelerated at first and he said he thought someone was following him.
"He was asked to provide a road side breath test and I asked him if he had had a drink. He said he had alcohol about an hour ago.
"Between being stopped and arriving at the police station he was asked why he was out at that time. He said he could not sleep and had gone for a drive."
The court heard how Allen was asked to provide another breath test at the police station, which was also positive, and he was subsequently charged.
As is customary in such cases, the DNA sample from Allen was taken and compared with the DNA profiles held on a national data base in relation to crime scene findings. His DNA matched the profile found on the body and clothes of Ms Class.
Allen was arrested at 7.24am on November 17, 1998 and interviewed by police.
Allen stood trial in January 2000 but was acquitted of the offence.
The jury previously heard how just a month after Allen had been acquitted, he approached two sex workers in Plymouth, where he had moved to, and requested oral sex. On both occasions he grabbed the women by their necks and squeezed them, demanding oral sex for free. He was convicted of both offences.
Allen is also alleged to have told an undercover police officer that he "strangled and dumped" a sex worker in the River Humber after she become angry and threatened to report him for rape if he didn’t give her some money.
Allen denies both counts of murder.
The trial continues.