You can reach out to the company and see if they are interested in buying you out. Depending on local laws (IANAL), to do that, they might have to issue a buyback. Smaller, privately held, companies probably won't do that as they lack the necessary cash.
Your next option is to do your own research. Start with the state of Delaware's division of corporations: https://icis.corp.delaware.gov/Ecorp/EntitySearch/NameSearch.aspx
Every company is required to file (depending on state) a list of officers and shareholders. That info should be publicly available. Reach out to those people and see if they are interested in buying your shares.
Another option is to find out when the next shareholder meeting is being held and let the company know you plan to attend. This way you can meet the other shareholders, and ask questions regarding the company's financial situation. That will help you get a better feel of what your shares are worth, as well as find a larger market for them.
Finally, ask the company for a shareholder list and financial statements. I don't know what the filing requirements are, but they should be providing atleast some of that information to shareholders annually. Signing an NDA prevents you from sharing the information but you can still hire an attorney or a broker to help you sell the shares so I don't see that as big deal. Your prospective buyer will already have this information, and if he or she doesn't ... well, that's to your advantage.