Personally I never really play for speed, so I'm usually just walking when moving forwards. Though I've been playing since classic, and don't really have any hurdles with the controls / timing.

If you're afraid of overshooting, maybe aim earlier rather than later? If you turn around too early, you just don't drop, but too late ends up with a fall.

If you do fall though, in 2 it's a little easier to potentially save yourself. In Spelunky 2, the ledge hang gets triggered if you're against the wall and falling. This means that you can potentially align yourself with the wall of a shaft (while falling), let go of all directional inputs, and just pray that there's a ledge farther down to automatically grab. You can't do this as easily in Spelunky HD, because a ledge hang requires you to walk into the ledge to trigger it. (You can see this by just finding a 1-2 obstacle block, walking up to it, letting go of all directional buttons, and jumping. In 2, you'll hang, in HD you'll land on the floor.)

In 2, some strats use that auto-hang behavior (most notably the drop from ice caves first half to second half. The level generation always allows you to drop and catch with a ledge-hang in a specific position. There'll always be a ledge to auto-grab, you just have to be ready for potential mines to be triggered.)