There are already a lot of good answers here, so instead I’ll offer you a game you might enjoy.

For context, I used to have the same problem, but I stumbled into a hack that turns it into a fun pastime rather than a nuisance or concern, and now it’s one of my favorite parts of taking up a new post.

What: Seed your own wild rumors by baiting known gossips with decoys.

How: While I’m often surprised by which rumors have legs, it seems prompt technique is the key controllable factor. Prompts that are more off-hand, indirect, and opaque seem to cultivate rumors with better proliferation and more colorful embellishments. I’m not sure why less is more here, but my theory is that the more gossips have to make up, the more they “own” their version of your story and the better they are at selling it. If necessary, you can drop some facts here and there (use only truths) but in general try to avoid details. Gossips are insanely good at filling these in on your behalf.

Who: This part is easiest. Gossips will always come to you. Usually they are some of the first people to approach you at your new job, looking for the scoop. The first you meet are usually the defacto gossip leads, but the rest reveal themselves quickly, especially if your rumors are good. You will learn to recognize them by the way they sidle up in the break room or pry opportunistically in conversation, however the easiest sign is that they respond to evasiveness and deflection in the opposite way others do.

Why:

1. Reconciling their conflicting versions of your personal story and private life disrupts their rumor mill internally and erodes their legitimacy externally.
2. You have the unique pleasure of receiving, over time, fragments of “your story.” This has utility for tracing the leaks and determining network composition, but mostly it’s just hilarious to see what they come up with, and interesting to know the versions of you that people will entertain.
3. As to social risk, so far none of my rumors have developed into anything terribly embarrassing — usually what they come up with is far more interesting than reality, sometimes even flattering — but the few times I’ve had to correct a story in conversation it was met with instant acceptance of the more realistic take. Most people know to take hearsay with a grain of salt.
4. I suppose the gossips might be mad if they realized you were toying with them, but I suspect they would just treat you as a dry well and leave you alone, which also solves your problem.
5. It’s relatively victimless as mischief goes. You don’t even have to lie. In fact it works better if you don’t.
6. And it requires almost no effort.

TL;DR: You can bait the nosy nellies into writing fan fiction for you. It’s easy, fun, and good for the environment. Your personality traits make you a natural at this game —others might need to tone down their instinct to overshare or practice being pointedly vague — but if in doubt just remember, storytelling is the gossips’ job and you are their humble yet beloved muse.