Andy Witton and partner Tracy Patel, of Innovation Way, were all set to take swab tests on Monday when the shocking discovery was made.
It was their 20-year-old daughter Stacy Ann, who is studying history at Newman University in Birmingham, that spotted the swabs had already been placed in solution, ready to be tested.
Councillor Stephen Craddock, portfolio holder for health and wellbeing, said he visited the family and apologised for the error and assured them there is no risk to their health.
It is believed a resident had taken the test and put the swab – secured in the container and in solution – back in the grey bag it arrived in rather than a box provided for used tests to be sent off to labs.
Hotel worker Tracy, aged 39, 47-year-old Andy – who works in a warehouse - and two of their children aged over 18 said the incident left them reluctant to do the test.
She said: “I opened the package and, as there were no instructions in the box, I messaged my daughter, who’s at university and who has to do the tests.
“She saw them and asked if we’d already done them. I said no and she said, ‘that’s already been done. It’s not supposed to come like that. It’s supposed to be flat pack and out of the solution’.
“At first, I didn’t believe it but when we opened the other packages, there was another one already done and three flat pack.
“Straight away, we moved them away, sanitised and got in touch with our MP because you don’t know who they belong to, what they’ve got or anything.
“I felt physically sick. I was livid. It’s the thought of the danger of it. You try to protect yourself and your family and you’ve got somebody knocking the door who could be giving the virus to you.
“You’d think that when they are delivering the tests, they’d be taking the completed ones away separately and not doing it together for this to happen.”
Since this happened, Tracy said two other people have come forward on her estate saying the same thing has happened to them. She added: "There's still people coming forward. We were told how it could have possibly happened but it's not acceptable with how severe it is.
"I've taken a new test and my daughter but my partner and my son are refusing now as they've lost all confidence in the system, which is understandable."
As well as contacting Walsall North MP Eddie Hughes, they also spoke to their old councillor and former town Mayor Pete Smith, who has raised the issue with senior members of the authority.
Mr Smith said: “It’s no wonder this family is upset because of what could have happened if two of them tried to use already used swabs.
“There is also understandable concern that this might be happening to others.
“We want answers as to how this happened and assurances that this has not happened before or will happen in the future. The family concerned deserve and require a full explanation.
“We want maximum uptake of both testing and the vaccination but occurrences like this are bound to test one’s confidence.”
Councillor Craddock said that changes have been made to the process to try to ensure it doesn’t happen again and provide reassurance to the public.
He said: “The used test kits had been placed in two clear bags and there was never a clinical risk to the family.
“I’ve been out to see them personally and spoken to them, hopefully reassuring them and apologised for the error. We have delivered them four new tests.”
He added: "We had one more incident reported to us which happened yesterday before the rebriefing.
"They come out in a grey plastic bag which is just bigger than A4, the kit is flat packed and there are full instructions.
"You make up a box, put the stuff in there and seal it and give us the box back.
"New tests are in grey plastic bags and used tests are in a white box.
"They've made sure that it won't happen again. They're checking the seals on packages as they go out and are dealing with the returns in a different way."