i don't think so, it rotates approximately every 25000 years, so the stars within it are doing pi*D and actually we are in the second last spiral of it, so escaping our galaxy probably should only take about 2000 years.
Discussion
The stars in our portion of the milky way make a full rotation around our massive black hole center approx once every 250 mil years.
You may be confusing with procession of the equinoxes which is a specific phenomenon to earth (and likely our entire solar system). Possibly the result of a binary orbit with a nearby “sister star” 💫