The Labour member said: “My partner came home and said 'I don't want you to do it anymore because the next time that phone goes, it could be a different conversation’.”
She added: “Totally shocked by what has happened to think that something so horrific could happen again to another MP, to another family. And scared and frightened – a real rollercoaster of emotions.”
Ms Cox was the last serving parliamentarian to be killed before Sir David on Friday.
She was stabbed and shot to death in her constituency – also Batley and Spen – while on her way to meet constituents at a surgery in Birstall in June 2016.
Speaking about Sir David, Ms Leadbeater said: “My phone started going straight away, my mum and dad, my partner, my friends, 'Are you OK?' and I was OK, I was visiting a school.
"But the shock and the feelings for us as a family, obviously what we went through and another family are going through that again, it's horrific. It's hard to put into words how that feels for me.
"The main people I am thinking about are David's family, his friends and the community he represents and has represented for such a long time.
"I think that's the thing that people need to understand, it's about a lot of people whose lives have changed forever today.
"People still come up to me and talk to me about where they were when Jo was killed. Their lives have just been changed."
She added: “At the heart of it are David's family and friends. I know for them now that their lives will never be the same again; they will think about this every single day for the rest of their lives.
"Even David's staff – so many other people today will have been out there trying to do the right thing, trying to do a really important job in public life, and this happens.
"I cannot believe that this has happened. It feels very raw for me.
"I know from messages I have received from politicians across the political spectrum, for them it is incredibly raw.”
Friday’s attack has now been declared a terrorist incident by police.
A 25-year-old man has been arrested on suspicion of murder and the investigation is now under the control of counter-terror detectives.