Adobe InCopy, as @HotButton HotButton said, is Adobe’s solution. I’ve been in several publishing situation where it’s worked reasonably well. The writers/editors make changes to the text in InCopy on their computer and those changes are reflected into the final InDesign document without you, the designer, needing to get involved. Everything is tied to a database, so it’s all reasonably seamless. The writers and editors can see in InCopy how their text changes affect the layout, as in whether or not the story is too long or two short.
Every publishing situation is different, and in some environments, like, say, technical documentation, where various specs are refined and updated over multiple versions, InCopy can be a huge time saver. In other situations, though, like a magazine or newspaper, InCopy can perpetuate the bad practice of writers being able to constantly meddle with what is supposedly finished copy.
If there are workflow problems in your environment (and from what you described, there are), InCopy can tend to institutionalize the bad practice of writers meddling in the editing, design and production. I’ve been in great, well-managed workflow publishing situations and bad, disorganized ones. A good workflow of writing > editing > design > production works extremely well when those sequential steps are adhered to and only overlap when genuine problems are identified. When that workflow gets scrambled with back-and-forth changes that ignore deadlines and the boundaries between the different steps, chaos always ensues and the end product suffers because of it.
So…, InCopy can either solve your problem or make it worse. The real problem I inferred from what you wrote is that the writers need to finish writing before you get their copy. Once you get the copy, their part should already be finished. The best situations have copy editors that serve as a buffer between the writers and designers, as @Eriskay Eriskay said. In other words, writers write, editors take what the writers have done, fine tune it for spelling, grammar, punctuation, etc., then pass off the finished copy to the designer for production. By the time the designer gets the final copy, it’s finished and any changes need to be negotiated with the copy editor and absolutely not with the writer.
If you’re in a situation where you’re working directly with writers instead of a copy editor, you’re likely working in a dysfunctional situation. I can’t even begin to stress the importance of a copy editing position in a publishing environment. If I were starting up a magazine, the very first person I would hire would be an editor/copy editor.