Will the at-fault driver's insurance company pay my medical bills?

No they will not. There are some circumstances where an at-fault driver's PIP insurance may pay some of your medical bills but that is the exception more than the rule. At-fault insurance companies will pay what a judge or jury makes them pay, which can include your medical expenses but only if the judge or jury awards those medical expenses.
Robert V. Clark
Maryland Car Accident and Personal Injury Lawyer