I don't follow the used market unless I'm looking for something collectible or no longer in production, so I'm no help in that respect. A new 3" Taurus 605 can be found in the mid $300. Bump it up $100 for a 4" Taurus 65.
357 caliber revolvers are not cheap these days. EAA Windicator sometimes sells for the mid $300..Not my choice but they do run. Even the used market is above your range. Good luck..If I were shopping the lowest I would go is Rossi or Taurus - Rossi RM66 or its 4" version $500, Taurus 65 or 66..$450 for $50 more a 627.
I was in this same boat a few years ago. You'll have to do some hunting but the right deal will come along eventually. I had the EAA Windicator, got it for $275 but mine was the snub nose version. A few months ago I found a Ruger Security Six with the 4 inch barrel for $475, traded in the EAA and got it for $365 out the door. If you wait the right deal will come along it just might take some time. I never thought I'd be in this camp but wait for a used Ruger or S&W you'll be way happier. The EAA did work never gave me any problems but my Ruger is a way better revolver. I would go no less than a Taurus but I'd inspect it really good first even if it's new. I didn't realize how bad the EAA was until I got my hands on a better revolver.
Rossi makes a 3" (no 4 inch) .357 that might fit into your price range. It's my understanding that Taurus' parent company owns Rossi. I think (not positive) that Rossi still makes the guns in its factory...someone else on here probably knows for sure. https://rossiusa.com/revolvers/rp63
Edit: Oops, your original post mentioned $300. As I read down through the various responses $400 got stuck in my head. You can probably disregard this post. Sorry, my mistake.
A bit of research says the $300 range is a tough place to be for a .357. Taurus, EAA, and Charter Arms are it. Personally I've never found a Ruger double action revolver with a trigger I liked. The SAA clone guns are fun, but again, a bit past $300 range. I think if I were you I'd look for a Taurus 605 if you really want to stay as close to $300 as possible, or a model 66 or 627.
I have a Charter arms 4” mag pug 5shot I use for yard work. I have a S&W 686plus 6” 7shot that is my most accurate handgun. One for about $350 the other for $700. For range use I’d get the largest revolver you can afford. Good luck Semper Fi and Can Do