Some ideas are:
- You branch off into another timeline and your actions make no difference to the previous timeline
- You’ve already taken said actions but just didn’t know about it so nothing changes
- Actions taken can have an effect (so you could suddenly erase yourself if you killed your parents)
- Only “nexus” or fixed events really matter, the timeline will sort itself out for minor changes
- something else entirely
Time travel wouldn’t be by jumps but by contionous change in perpendicular* time dimension.
So earth wouldn’t escape from your feet as you would move with it, just like you are doing right now with just one dimension.
*time travel would be imposible if you can move only in positive direction.
Then the 2nd time dimension would need to be under some funky angle (3/4π>α>1/2π and α≠π and α≠0)i am wrongEdit: After some thought, to truly time travel the second dimension would need to be parallel to the original one but backwards. So some people would be living in reverse time. (I have seen that concept in Sci-Fi)
Still the perpendicular time dimension is too funky of a concept to truly give it up.