How many is "quite a few stops" in 15 years?
If you've been at, let's say, five or more stops during that time, there's pretty much no way of attesting that you offer some long-term stability. Unless, of course, you've been laid off multiple times in those 15 years ... that's a different story.
Your cover letter will have to do some of the heavy lifting. You'll certainly want to explain being laid off in your most recent job(s). You'll also want to explain/spin why you've changed jobs so often (especially if you've moved to positions of higher responsibility or authority) if you think you've moved a lot.
That said, journalism is a volatile industry and people don't expect you to be a lifer anymore. (It's the same in the real world, too.) As recently as 10 years ago, there was a belief that people would have seven jobs in their lifetime. However, technology and the economy have changed that game considerably, so it's not unreasonable to have more jobs than that these days. It is, however, entirely incumbent upon you to showcase yourself as a dependable employee -- someone with the desire to stay put for a while -- during your phone or in-person job interview, and reassure your prospective employer of that fact.
Good luck.