[ShinyRice]

> If you're traveling at 10 m/s as you go out of the gravity well you go faster and faster because time speeds up, but your velocity per unit time stays the same.

You seem to be confusing coordinate time with proper time. The time you experience as you get out of a gravity well is proper time, which, as a sibiling comment said, is something you obtain when you do the path integral of the path you travel. This time is not exactly affected by dilation. Anything traveling with you will not have its time go faster or slower. Anything that is not traveling with you, say stationary on Earth, will indeed have its passage of time shifted with respect to you.

Coordinate time is the time observed by an observer at the center of the gravity well, looking at you (not always, but generally so for most calculations. Also, observer = origin of frame of reference) and that is what is morphed and bent around. Since you're an observer as well, you also see other points experience time differently, since from your point of view, they're moving away from you.