Yes, make copies and put the original aside. Then take the unsigned copies with you when you go to gun shows. I always sign in blue ink so there's no question that it's an "original" and not a copy. You can also mail these copies to vendors when you need to use them.
If you have a scanner then scan a signed copy of your license and keep it on your hard drive. Some companies, like AIM, will accept an email with scanned .jpg image or faxed copy but others will prefer that you mail it in.
A side benefit to your C&R license is that you can get dealer pricing from some companies. Midway, Brownell's, and Southern Ohio Gun are three that come to mind. There are others, as well.
-Bob