film industry - What role do screenwriters play, after handing in the script?

Their role is to get their next gig, stay off the set. There are exceptions, of course. No one would ask Aaron Sorkin to leave the stage. But if you just wrote your first 800K horror movie, it's unlikely, but not impossible, that a new writer would be asked for much more than that. That said, directors who know they like reading but not writing, often will ask the writer to visit to help with dialog, especially with dialog, when circumstances require. Powerful stars will also, under some circumstances ask (demand) that a writer be present to help with situations after its clear no one on the company, including the director, can write quickly and effectively. I've worked with directors of some standing who are incredibly ill-suited to oral communication. Once can only guess at the consequences of an actor saying "I can't say this line. It makes no sense. I don't understand it. My character would NEVER say this." Without trained writing talent ready to go, on set, in the conversation, a delay is inevitable. Delays cost money, shots made, shooting light and other perishables. So then the writer is there for protection. Doesn't happen often, but does happen.