Praise be to Allah.
Firstly: Prohibition on commercial insurance
But if someone is forced to have insurance, there is no blame on him, and the sin is on those who compelled him to do that.
Secondly: The insurance company has no right to increase premiums during the contract period.
Insurance contracts are of limited duration, and are renewed at the request of the one who is insured. The company does not have the right to increase the amount of premiums during the contract period, but it may do that when the contract is renewed, and the person who is insured has the option to cancel.
If the duration of the insurance contract is one year, for example, and premiums are paid monthly, the company has no right to increase the premiums during the year of the contract. If it does that, and you are able to leave the company, it is permissible for you to do that.
It is not valid to increase premiums because the insured person caused an accident, because the insurance contract is based on that; the company covers accidents, pays the costs of repair and keeps its margin, so what is new if the insured person has an accident?
If premiums are increased during the contract period, the insured person does not have to accept that. If he is forced to accept it, but can find a way to leave the company, it is permissible to leave it in order to protect his wealth.
If the company states in the contract that it will increase premiums during this period if the insured person causes an accident, or if costs of repair exceed a certain amount, then this is ambiguity in addition to what the insurance contract contains of ambiguity and riba, and the insured person is not obliged to fulfil this condition. He may leave the company after the accident if he is able to do so, because this is an invalid condition in an invalid contract that he only entered into because he was compelled to do so.
And Allah knows best.