The insurance adjuster for the other driver told me that she won't pay my medical bills while I am receiving treatment, and that I will have to wait until I settle my case to get reimbursed for my medical care. Is this true?

For the most part, this is true. Most liability insurance carriers only pay the other driver's medical bills at the time of final settlement. They do not pay medical bills as you go along. You will need to pay your own medical bills (either with your own insurance or with your own money) while your case is pending. If the accident is truly the other driver's fault, you should be reimbursed for these expenses at the time of settlement or, if the case is not settled, if you win at trial.