The answer generally is yes to both. Your medical claims should be filed against your auto insurance first. If you reach your coverage limits for your auto insurance medical coverage, you can then file against your health insurance, subject to any deductible. If you have sufficient medical coverage under your auto insurance policy, you might not have to pay anything for your medical bills. However, if your medical bills exceed your auto coverage limits, anything you submit to your health insurance company will be subject to the deductible, which is the amount you have to pay yourself before the insurance company begins to pay.
The amount of your deductible depends upon your policy. Also, your health insurance policy likely has a subrogation clause, meaning if you recover damages for your injuries from another party, either the person responsible for your injuries or their insurance company, you must reimburse your health insurance company from that settlement for claims paid by the health insurance company. If you put your health and auto insurance companies in contact with each other regarding your injury claims, they can work together to sort out who is paying for which claims and also settle any subrogation issues.