This reminds me of the games you can play flipping signs in relativistic equations when you make v > c. You can interpret those in a lot of fun ways too.
Is that because you could move away from the center (a bit) to delay your progress toward the center singular?
You can’t move away from the center. You can move in direction around it (while still moving towards it at the same time). And you can move in the previously time now just space.
Delaying your fall is precisely the same as trying to avoid tomorrow by moving around. Moving away from the center is like moving back in time. No matter how you move in the spatial dimensions you can’t do it.
With the exception of faster than light travel. That allows you to move away from the center, and it also allows you to avoid tomorrow. It’s pretty much the same as time traveling in general relativity.
How it happens? Normal spacial direction has positive sign. Time has negative. When crossing the horizon the equations become such that t gets a positive sign and the radial direction a negative one. After that they behave as expected.