If it is a top break S&W in .44 with a square butt, it is either a No. 3 or the "old, old" Russian. The latter is marked "Russian Model" in either English or Russian. In the absence of those markings, it is the No.3 and is chambered for .44 American, a smaller diameter cartridge than the .44 Russian (and .44 Special/Magnum). A few were chambered for .44 Henry, but those are obvious as they have the rimfire firing pin. I don't think anyone is making S&W American ammo, but the "cowboy" ammo companies are always adding new cartridges, so I am not sure.
The No. 3 First Model went to s/n 8000, the Second Model from there up, so your four digit could be either. There was a transitional model in the 6700-8000 range.
As with all other antique guns, condition is everything for value. In like-new shape (very unlikely) values can go to $7-8000. In 60%, around $2500. Guns in .44 Henry bring a premium as do the rare guns marked U.S. or Nashville Police. The latter is so rare there is no price basis.
A way to double check caliber is to see if a .44 Special or .44 Magnum is too large in diameter to fit the chamber. If so, the gun is chambered for .44 American and is the standard No.3.
(NOTE: No flames, please; I am NOT suggesting trying to fire a .44 Magnum in an S&W No. 3, only use it to check the chamber diameter.)
Jim
