I have had this before… how to conduct yourself may depend a little on what your job is. For example, psychologists/social workers may have more trouble because they need to “join” (jargon term)with the clients to be able to help them.
Some people really understand and think highly of you for it- just say “for religious reasons I don’t shake hands with people of the opposite gender except for immediate family and spouses”. I had someone sincerely ask me what religion I represented and they were completely fine with it. They might ask you a question and just respond truthfully without making yourself sound “wacko” or “holier than thou”.
Sometimes you have a feeling you won’t be understood. In that case hold a resume in that hand (good to be prepared with that as well). You might want to hold a tissue in your hand and apologize for being under the weather. This works well if you are on an interview or first day on the job, when you may have to shake many hands… I have had this experience before and found being honest, sincere, but not aloof works best! good luck- it can be uncomfortable at first until you practice your “speech”.