There is no such cartridge as ".44 S&W Schofield". The Schofield model was chambered for .45 Revolver Ball, which we now call .45 Schofield. It was basically the .45 Colt (aka .45 Long Colt, or .45 LC), with a slightly larger rim, and a cartridge case that was about 0.10" shorter than the Colt's cartridge.
The fact that you have the patent dates on either side of the barrel
does indicate it is probably a Schofield. It should say so on the LEFT side of the barrel. If the locking latch is attached to the frame, rather than the top strap, and the grip is a smooth curve, rather than having any sort of bump, hump or knuckle on the backstrap of the gripframe, then it is likely a Schofield. What barrel length is it? If 7-inch from the front face of the cylinder to the muzzle, that is correct for an original Schofield.
If the side of the latch has a little round knurled boss on each side, with the rear sight rounded, then the gun is a 2nd Model Schofield, and with the serial number in the #59XX range, it should be! As to the value of the gun, it depends on the condition. In good shape, with a lot of finish and mechanically tight and sound, I would guess the price quoted is in the range. But I would contact some of the known dealers and see what they estimate it. It depends on how badly you need to sell it, of course. Also, do you have any family history to go with it? That might interest someone in addition to its intrinsic value. Sorry, I am NOT in the market.
Hope this helps.
Oops! The patent dates are on the LEFT side of the barrel!