It's a .38 Single Single Action, Second Model. With the longer barrel and target-type front sight (if it is original), it's a less common configuration.
I think the grips have been changed. The .38 Single Actions typically had a different checkering pattern and block letter S&W on their hard rubber grips.
I don't think it would cost much to put it back in shooting condition. The good thing is that cylinders from several of the double action models should fit (although they won't match the serial number). The buggered trigger screw and mainspring strain screw can be swapped or fixed. I would suggest sending it to Dave Chicoine
www.oldwestgunsmith.com for evaluation and repair. He's the master. He's in NC. The good thing is that shipping it to him won't cost an arm and leg. It's a bona fide antique and can be shipped through the post office by any individual. It is no longer considered a firearm. Good luck!