This is the Colt New Army & Navy.
The basic gun was made from 1889 to 1908 under various model numbers.
Your chances of finding a good barrel are slim, and an original nickel finish 5" usable barrel are astronomical.
Even if you could find a usable barrel, you're going to find it just about impossible to find any gunsmith who has the special tools needed to change out the barrel.
The old trick of using a hammer handle through the frame as a "wrench" WILL destroy the gun.
I simply don't have any good news for you.
These guns are 100 years out of production and the supply of parts in good shape have long since dried up.
Very few were produced in nickel, and I doubt a good barrel in nickel could be found at any price, much less a 5".
Last, these were very complicated actions that easily break or get out of order.
These days, these are pretty much considered to be collectors display guns.
My advice is to retire it as a family heirloom. Your chances of getting it re-barreled are almost impossible, and the chances of a local gunsmith doing it without ruining the gun are equally high.