“Do I have to give a recorded statement to the at-fault driver’s insurance company?”

No, you don't have to.  In fact, it is always a bad idea because whatever you say will be manipulated by the insurance company to maximize their chances of paying you as little as possible for your injuries.  Of course, they want nothing more than to deny your claim entirely.  Therefore, do not play into their hand, and refuse to give a recorded statement to the insurance company that insures the driver who caused your personal injuries.

Also, the adjuster will be nice to you in order to befriend you, and to convince you that they just want to be fair with you.  Nothing could be further from the truth.  The adjuster wants to get rid of you with as little money as possible, before you get an attorney.