Hello Randy1944,
The Sharps Factory Records do survive they belong to :-Dr. Richard Labowski
4613 Osage Ave.
Philadelphia Pa. 19143
Phone No 215-748-1376.

The factory didn't record the built date only the date it was dispatched. You can have it "lettered" by Dr Labowski but that is I understand in the region of US$ 200 possibly more. If you ring him he may tell you the details over the phone, so have a pen and paper to hand before you ring him.

The most likely recipient would be the US Army. If so, there should be inspectors stamps in the carbine and possibly other military markings. Sharps was the second largest supplier or rifles and carbines to the US Military during the War of Northern Aggression, only the Spencer repeating rifles were made in larger numbers.
Good luck with your request. and welcome to the Forum.

Harry

Edit. Some 60,000 of the 1863 Carbines were made and numbered 71,235 to 99,999 and c1 to c 49,528.
Production ran from 1863-1865. A few were made with a coffee mill built into the butt of the carbine. Stock was Walnut and the action, block and other parts were Colour Case Hardened only the barrel and sights were blued.

The Rifle version serials ran from c 30,000 to c 40,000. Only made during 1865, total production was 7,000. Another rifle version (New model 1863 sporting rifle) was made during 1865 for commercial sale but was also taken by the Government for war use. These are the rarest of the Sharps 1863 models.

Harry

Hello Randy,
I would suggest that you hang on to the original screws like a drowning man clutches at a few straws. It's likely only the heads are 'boogered up' and not the threads. I would suggest that if you have the skills that the peen the displaced metal gently back into shape, if that cannot be done then get a gunsmith to weld them up with a soft steel and recut the screwdriver slot. IMO it is better to preserve the original items than replace them if you can find anyone who can duplicate the threads. Remember these were done long before thread standardisation, was introduced, so modern threads are unlikely to work. You may find someone who can duplicate the originals, but they won't be cheap. If you can get the originals fixed then have them case hardened, and buy screwdrivers that fit the slots perfectly, is use of incorrect sizes of screwdriver that caused the problems in the first place. Personally I make my own and have done for every rifle or handgun I have ever owned. I use Silver Steel but in the USA Drill Rod would be ideal. They can be made in minutes on a milling machine. Heat treat the tip to a straw yellow colour and quench. 

However, don't despair, another member may actually make them, so put an add in the Wanted and For Sale Forum on this site, or someone may know of someone who does make them. If you don't ask you won't find out. Good luck. As for loving the Sharps, all I can say is you have very good taste Sir. Photographs of your rifle would be appreciated by all. Hint, hint.  
Harry