Coulson was last seen near a church meeting she had attended on Raglan Street, Coventry, on 21 May, where she had appeared to be well.
Durnion is believed to have met by chance his victim, who was a sex worker, shortly after she left the support group.
He told police he had asked her to buy drugs for him and there was no sexual contact between them.
Investigators said she went with Durnion to his flat, near the city centre, just after 23:20 BST.
Later that night, police officers were called by Durnion’s neighbours, who reported hearing a woman “screaming in fear” inside his flat.
When officers responded, Durnion would not let them in and told them he was having a mental health crisis, pretending his father had cancer.
But he had killed Coulson just minutes earlier, and hidden her body underneath a mattress and duvet.