What is the best way to begin searching for attorneys to assess my case and represent me in a suit against a hospital/doctor?

It absolutely is a reasonable concern.  The good news is that the lawyers who represent the doctors and hospitals generally don't seek out cases like yours because the insurance companies who hire them to defend the doctors and hospitals won't let them.  There are a couple of ways you can do this.  You can look for lawyers in your area who belong to certain groups like the American College of Trial Lawyers (www.actl.com) or the American Board of Trial Advocates (www.abota.com).  You can't buy your way into these organizations; you have to be invited and the lawyers who are invited are typically pretty high quality.  These organization invite both plaintiffs lawyers (that's the kind that you want) and defense lawyers (who represent doctors and hospitals) so you'll have to look at their websites to be sure that you're calling a plaintiff's lawyer.  Another option is to check out SuperLawyers and search for plaintiff's medical malpractice in North Carolina. I wish you the best of luck.