How long are you going between cleanings?
If the .357s are difficult to chamber, it definitely needs to be cleaned more. It's the heavy build-up of crud that can potentially cause a problem. Shoot 100 lead .38s, then shine a light into the chambers. Those standing rings are a potential problem... I doubt they would survive a bore snake, although I usually just brush them out with FP-10 (a CLP, but Hoppes 9 or whatever is fine) and a .40 cal brush. At the range, I'll shoot .38s and end a session with .357s without even the brushing. It's still hot, hasn't really set and isn't that built up. It's when you don't clean your gun for multiple sessions, let it set and harden and built up, then push your luck with tightly chambered .357s. In general, if you can drop the .357 round in, you can shoot it, but in general it sounds like you should be cleaning your gun more.
I clean my reolver every session. If I want to do a half-assed job, I can do it in 15 minutes. Usually it's 30-40, sometimes an hour if I've put 400-500 rounds down. It's just some spraying, brushing, jagging, wet and then dry-patching. It is redundant and a bit boring, but a) there is nothing like showing up to the range every time with a gun that looks like it's never been fired and b) you eliminate all these problems.
Before you fire next, take a patch with a few drops of CLP, Breakfree, or my fav FP-10 and put a light layer on the forcing cone, front of the cylinder and back plate. When you are done shooting, take a few dry patches and you can wipe 90% of the black crap off. And if you wait until you get home to clean, it still makes it much easier. Learn to love the FP-10. It is your gun's best friend.