Do I have to talk to the other driver's insurance adjuster?
You have a duty to cooperate with your own insurer under your policy. You owe nothing to the other driver's insurance company.
They will call quickly, often within a day or two, and they will be friendly. That's not deception — it's their job. But the adjuster's function is to evaluate and limit the company's exposure, and everything you say serves that evaluation.
What goes wrong. You describe injuries before you know what they are. You estimate speed or distance and get it wrong. You say you're feeling better because you're being polite. Each of those becomes a fact in the file.
What to say instead: confirm the crash occurred, provide your name and contact information, and decline to give a statement or discuss injuries until you've spoken with counsel. That's complete and it's not obstruction.
What you should do: notify your own insurer promptly, since your policy requires it.
Give them your name. Give them nothing else yet. ---
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