If I am grappled by an enemy with 10' + reach, how does our movement work?
For most grapplers, since as you quoted,
you can drag or carry the grappled creature with you, but your speed is halved
Moving while dragging or carrying the grappled creature is explicitly stated.
Can the grappler move you back and forth within their reach?
Are you stuck at the same distance that you were at when they initially grappled you?
There isn't an explicit method to move the grappled creature closer without moving, unless you have an ability like a Roper's Reel.
Reel. The roper pulls each creature grappled by it up to 25 feet straight toward it.
Are you stuck at the same distance that you were at when they initially grappled you?
Since it says you can drag or carry, the grappler could opt to move without moving the grappled creature, as long as they stay within reach and continue the grapple.
This presents the option of the grappler moving closer to the grappled creature without moving them. This is probably a pretty common tactic for a creature with grappling tentacles with a reach greater than 5' and a bite with reach 5', but without Reel.
Can they only move you closer/further when they move?
Since you can drag or carry the grappled creature, you could move without dragging or carrying the grappled creature, and then move while dragging or carrying the grappled creature. For example, the grappler moves to within 5' of the grappled creature, then moves away, dragging or carrying them, keeping the grappled creature in an adjacent square.
You could move them without moving by using Improvising an Action
Page 193, Player's Handbook introduces Improvising an Action
Your character can do things not covered by the actions in this section...the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure
For a grappling creature to pull a grappled creature 5', this DM would just adjudicate it as a grapple,
If you're able to make multiple attacks with the Attack action, this attack replaces one of them... a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use).
On success, the grappled creature is moved 5' toward the grappler.