Margaret Jones, a resident at Oaktree Court, in Wellington, Somerset, for eight years, has had to find alternative accommodation in Taunton.
Care provider Majesticare said it was unable to comment on the case.
Somerset County Council pays up to £575 per week, per resident but said it could not fund the increased fees.
The situation at Oaktree Court was came to light last month.
Mary Pike, Mrs Jones's mother, said the move would be "very emotional".
"We shouldn't be going through all this now. She'll be 94 [soon]," she added.
"[The move] is going to be tell-tale really about how she's going to cope. She's leaving her home that she's known for eight years."
Mrs Jones told the BBC she "didn't like the move" and had been "happy" at Oaktree Court, eight-miles from her new location.
Wellington councillor, Labour's Andrew Govier, said: "With an ageing population, it's really difficult.
"But the council is really struggling. They're not getting money from central government so it's very difficult for them to keep upping the fees."
BBC Points West's Clinton Rogers said a dozen other residents on county council contracts had also been affected by the increase in costs at Oaktree Court, which came in to force on 1 January.
Six are moving out; four are staying at the new rate; one person is too ill to move; and two residents are, as yet, undecided.
Somerset County Council funds 1,500 care home beds across the county and spends a weekly average of £575 per person for nursing care and £444 for more straightforward residential care.