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I think it's easier if you imagine space/time like a flat plane that dips depending on how much mass the central object (like the sun for example) has. Earth circles around that dip much like your drain plug analogy. If the sun disappears, it still takes time for that dip to rise back up to a flat surface. That's the speed of gravity.
As soon as space/time begins to flatten beneath the earth, its momentum begins to turn into a straight line, rather than an orbit.