H & R Model 04 .32 cal, Family Heirloom, Questions

I just got/inherited a nice little H & R Model 04 in .32 cal. D/A, nickel plated 2.5" barrel. It was my Grandfathers before it became my Uncles, before it became my Dads, and now becomes mine. It came with a box of .32 S&W ammo, very old and crusty looking stuff. Looking in all my reloading books, I only find .32 S&W long, nothing designating short, or just plain .32 S&W. The box of ammo I got fits in the cylinder,  but is quite short. the brass is like .660 long. The box the ammo came in says it is .38 S&W, and is definitely not the correct box. The cylinder is certainly long enough for the .32 S&W longs, I am just not certain that its a good Idea to run them. Is/Was the .32 like the .38 spl., and the .357 where you can use either one?

I spent most of today cleaning it up, getting the Grunge out of it with my trusty can of Kroil. The barrel cleaned up unexpectedly well with a high shine . the cylinder came out a little worse as somebody had left it loaded a while, and there is staining in the chambers.

Also, I have not fired it yet as the trigger does not reset forward, and one of the grips is cracked at the alignment pin. Does anybody have suggestions on where to look for parts, and or a mechanical schematic of this gun? I really want to make it as close to 100% as possible given it has been in my family roughly 50 years. I don't  want to spend a fortune as this was considered a utilitarian type of gun as opposed to a "showpiece". I guess given the condition its in, I would really like to be able to shoot it every now and again.