Local authorities are being pushed "perilously close to the financial edge" and most will have to continue to divert funds into social care services "at breaking point", the Local Government Association (LGA) said.
Failing to stump up the money will continue to "heap further pressure and wasted expense on the NHS", said Lord Porter, the body's Tory chairman.
The Government said funding will exceed the £5.8 billion gap the LGA estimates will be faced by councils over the next four years.
All councils can raise taxes by up to 1.99% in April, while those responsible for social care can increase bills by up to 3% more.
Nearly all local authorities responsible for social care have approved or are considering applying the precept, the body revealed.
The extra cash must be spent on social care, however the LGA warned that the money will be "swallowed up" by the burden on councils paying the National Living Wage.
The continued shortfall, estimated to hit £2.6 billion by 2020, means other services like bin collections, running children's centres and libraries, filling potholes and maintaining parks will miss out.
The LGA has urged the Government to provide the extra funding for social care when it comes to the Commons with its final Local Government Finance Settlement later this month.
Lord Porter said: "Services supporting the most vulnerable people in our communities are at breaking point and many councils are increasingly unable to turn down the chance to raise desperately-needed money for social care and other local services next year.
"But extra council tax income will not bring in anywhere near enough money to alleviate the growing pressure on social care both now and in the future and the social care precept raises different amounts of money in different parts of the country.
"Social care faces a funding gap of at least £2.6 billion by 2020. It cannot be left to council taxpayers alone to try and fix this crisis.
"Without genuinely new additional Government funding for social care, vulnerable people face an ever uncertain future where they might no longer receive the dignified care and support they deserve.
"This is not only worse for our loved ones but will also heap further pressure and wasted expense on the NHS."
From April all councils can raise council tax by up to a threshold of 1.99% without needing to hold a referendum, while most district councils can increase bills by £5 for Band D homes.
Some 147 of England's 151 social care authorities are either considering or have approved the precept for 2017/18, the LGA said.
So far 108 councils, around three quarters, are considering or have approved introducing the full 3% rise, while a quarter are considering or have approved a 2% rise.
It leaves just four councils who will have announced they will not apply the social care precept and they intend to, or have, frozen council tax next year.
The Government has said it will prop up services with a £240 million adult social care grant, but the LGA argues this is simply repackaging funding originally earmarked for house building.
A Department for Communities and Local Government spokesman said: "Whilst local authorities - like all public bodies - have had to find efficiency savings, our historic four-year funding settlement gives them the certainty they need to plan ahead with almost £200 billion available to provide the services that local people want.
"By the end of this Parliament, councils will be able to keep 100% of local taxes.
"We've also announced an additional £900 million for social care meaning councils will have £7.6 billion of dedicated funding to spend over the four years."