If you started heading due east or due west, and went in a straight line you would eventually be heading south.
If instead you continued heading due east or west, you would travel in a circle
At a radius of 6371 km from the north pole, this would amount to turning 1 degree per ~111 km.
You would not be able to notice this.
You would need extremely precise tools to detect this.
Likewise, on the RE, if you were to travel due east anywhere other than the equator, you will be going in a circle turning in 2 directions. You would be turning down to stay on the surface of the round Earth rather than going off into space, and turning left or right to stay heading due east.
Why do you have to "continually make turns" on a globe?? East is always east West is always West.
While east is always east, the direction of east is not always the same.
If you were just following the surface, if you were north of the equator you would end up heading towards the south, crossing the equator and then heading east again and then heading north and so on.
For a simple example, consider an area near the pole, where it works just like a hypothetical flat Earth.