I haven't seen this N frame .357, but if it were mine, first thing would be to count how many coils the rebound slide spring has. Less than 17, I'd replace it.
The other thing I would look at is to carefully stone the bevel of the trigger to sharpen it. This will increase the single action trigger pull, and eliminate the push off condition as described.
Unless some serious ill fated filing has been done previously, I don't think it would require any new parts.