Welcome to the forums from the Wiregrass! You have a .32 Military & Police, Model 1902, 1st Change from around 1904. The .32 Winchester is also known as the .32 WCF (Winchester Center Fire) and .32-20. These are all the same cartridge. Your gun has an older action that is not compatible with anything made after 1905. So take care to keep it clean and lubricated and shoot commercial ammo if you can find any. Most folks that shoot these are handloaders. Cowboy ammo is available but you will likely have to order it over the internet and wait...wait...wait.
It's going to be hard to find parts for that gun. You might check with Numrich, but putting ads in the classifieds at the S&W forums might be the best bet. Yes the strain screw needs to be screwed in all the way in order for the action to function properly.