Should I speak with a lawyer before cooperating with the other party's insurance company?

Definitely yes, because the auto insurance, they represent the other side and so they don't have your best interests at heart. Presumably, you have not been in a car accident before so you really don't know what to do so you definitely want to talk to a lawyer to find out what you should be cooperating on, what you don't have to cooperate on because the other insurance company isn't going to be the one to volunteer this information to you because their best interests is with their insured, with the other driver. Definitely reach out for an attorney just to even ask some questions because it will help you in the end.