Families who rely on state help to pay their rent will have to find hundreds of pounds extra every month to to avoid eviction because of the government’s freeze on housing benefit, the homeless charity Shelter has warned.

Demands from Shelter and other housing organisations for a government rethink come as a number of local authorities – mostly in the south of England – report that many of those being evicted and rendered homeless are now in full-time work, typically in public sector jobs such as teaching or nursing.

Ahead of this week’s budget, the charities are calling on ministers to end the freeze, which is set to run until 2020, and to build more affordable homes in areas of need, or to risk a further explosion in the already rapidly rising numbers being placed in costly emergency accommodation by their local authorities.

One single mother of two young children in north London who works full time as a teacher told the Observer she had always thought her job, and contribution to the community, would ensure she could find a respectable home. Instead, after her partner left and her landlord put up the rent to a level she could not afford on her own, she was evicted. She is now living at a B&B in east London where her children have no room to play.

“It is really sad that there is nothing available for someone in my position,” she said. “I genuinely thought that being a teacher I would be OK. To think that the government cannot do better for me and my children is really terrible.”

The new research by Shelter has found that inone in four areas of England, the combination of rising rents and the benefit freeze means families with one or two children living in a two-bedroom rented home and claiming housing benefit will have to find at least £100 extra rent a month over the next year.

In many parts of southern England the shortfalls will be more than £200 a month. In London they will rise in areas such as Tower Hamlets, Greenwich and Hackney to nearer £400 a month. In Camden, the shortfall will be over £600 a month, rising to £1,252 in Kensington and Chelsea.

The housing benefit cap was first announced by George Osborne as part of a drive to cut £10bn from the benefits bill. But the policy is backfiring, and hitting those in work and landing local authorities with the bill for emergency housing for the homeless.

The number of households that have become homeless after an eviction over the past year is up 12% compared with a year ago at 18,820 while the total number of households in temporary accommodation has risen to 74,630, up 9% on a year earlier. Eviction by private landlord is now the most common cause of homelessness.

Roger Harding, director of communications, policy and campaigns at Shelter, told the Observer: “We have grave concerns that the current freeze on housing benefit is pushing hundreds of thousands of private renters perilously close to breaking point at a time when homelessness is rising. For those hit by the freeze, housing benefit is failing to bridge the widening gap between escalating private rents and incomes that simply can’t keep up.

“Whether you’re a struggling family, or a young person on a low-paid zero-hour contract, in many areas of the country you’ll face a grim uphill battle to keep a roof over your head and food on the table. And if you have to move, you’ll find it extremely tough to find a landlord prepared to accept a rent you can afford.

“With next week’s spring budget looming large, we’re calling on the government to abandon the freeze in the short term or risk making more people homeless. Ultimately, if the government wants to cut the welfare bill in the long term, they should concentrate their efforts on building genuinely affordable homes that low earners can actually afford.”

Shelter calculated the maximum amount of local housing allowance (the name for housing benefit claimed by private renters) that households would be eligible for in 2017-18, and then set this against the cost of renting in different parts of the country to establish the deficits people would face.

The alarm has also been sounded by the Local Government Association in its official submission to government ahead of the budget. It points out that the failure to build sufficient affordable homes over the last few decades has driven ever more people into expensive rented accommodation, which has to be topped up with housing benefit. When renters find they can no longer meet the costs, the local authorities are statutorily obliged to find them emergency accommodation.

The LGA submission says: “It is local government that is picking up the pieces from the long-term failure to build homes that are affordable for families … placing more families in expensive temporary accommodation [and] costing councils £3.5bn in the last five years, rising 43% in that time.”

In many parts of London the problem has reached “breaking point”, according to the mayor of Hackney, Philip Glanville. In his area, the average price of a home has increased by 82% in the past five years, with rises higher than anywhere in the country in the past 20 years. Private rent now averages £1,800 a month for a two-bedroom flat, a 34% increase since 2011. Shelter calculates that in Hackney Hackney families renting a two-bedroom home and claiming housing benefit will face a shortfall of £383.51 a month in 2017/18 as a result of rent rises and benefit freezes.

Glanville said: “This means there will be more people who cannot find the extra and who are at risk of being evicted. People assume that it is the unemployed who end up in emergency accommodation but it is so much broader than that. We are finding it is people who are in in work, who are striving and key jobs, such as teachers, nurses and trainee police officers. I don’t know how the government expects people to go out to work, take their children to school and then return to one room.”

Today, in a letter to the Observer, the heads of 15 charities and other organisations representing vulnerable people with mental health and other conditions that mean they live in “supported housing” – whose rental costs are currently covered by housing benefit – voice their concern about the impact of proposed cuts on the most needy.

“Such housing provides safe and independent long- or short-term support for people with learning disabilities, women and children escaping domestic violence, older people, people living with mental illness, people who are homeless and others,” they say.

“Until now, they could rely on housing benefit to cover their rent and some service charges. Yet, under government plans, their benefit will be capped by an arbitrary formula and most will need a top-up from their local authority to fund this crucial help. As local councils already face vast budget cuts, this will cause problems and uncertainty. The current funding system is not perfect, but the proposed system jeopardises the wellbeing and safety of many vulnerable people – the human and financial costs could be great.

“This can only spell bad news for our NHS, care and other public services left to pick up the pieces. We urge the government to give the people who rely on supported housing a guarantee that they can access this housing in the future.This requires a new fair funding system that recognises the higher cost of this housing and is built on the actual needs of vulnerable people.”

Jon Sparkes, chief executive of Crisis, said: “Homelessness in England has risen for the sixth year running, while temporary accommodation and B&B placements are up 52% and 250% respectively since 2009/10.

“We need housing benefit that actually covers the cost of renting, a stronger focus on preventing homelessness, and more support made available to homeless people trying to find a home to rent. At the same time, we need decisive action to make renting more accessible and affordable, along with radical solutions to tackle the severe shortage of truly affordable homes.”

The Department for Work and Pensions said: “The changes to housing benefit are restoring fairness and ensuring there are no perverse disincentives that trap people on benefits. But we understand that there are some people who will need additional support. That is why we are providing £870m in discretionary housing payments to support those who need it the most. Even with the reforms, we continue to spend £24bn a year on housing benefit.”