It doesn't require that a Dyson Sphere be an isotropic radiator, though. It could be emitting its heat in a different direction.

Well, if you emitted it all in a beam away from the plane of your galaxy, that might make you very stealthy. That would impart thrust on the sphere... which if it wasn't perfectly balanced might cause it to wobble, and then start rotating...

... and now we have a sci-fi reason for quasars. :)

You could direct the waste in two beams perpendicular to the plane of the galaxy, in order to balance net thrust.

I question how stealthy that would really make you, though. If someone was bothering to look for stealthy civilizations, they'd just have to scatter some observatories around their region of the galaxy, then watch. Eventually, your structure would occult a star from the POV of one of those observatories, or they'd pick up your beam scattering in interstellar gasses or dust clouds.

Of course you'd need two beams. If you used a single beam it would be a stellar rocket... which might be fun in different ways. Certainly gives the phrase "interstellar warfare" a different meaning.

I realize this is all space opera, not hard SF. It's still fun.

"But Sir, these are the correct stellar coordinates for the Zorgoph home world! They should be right here!"

"There's nothing here, Captain. Nothing at all. So you're saying that they moved an entire star system? In the last solar cycle?"

Good times, good times.

Maybe they figured out how to create an artificial wormhole and direct their waste heat into that....

buzzkill ;)