Hyperion, that's close except for the apparent physical clustering of star systems.
You wouldn't actually need that in a universe where the medium-level (systems connected to each other through wormholes) structure is defined not by physical proximity of systems but their connective relationships to each other as nodes. System A and System B might be directly connected, and so would appear next to each other on a node map, but their locations on the string could be immensely farther apart than System A and System C if System C were on a nearby but separate string.
This can work as long as travel time is invariant between any two systems on a string regardless of distance between those systems on the string.
And Cornflakes, that is some nice theorycrafting. I like it: "webspace."
A thought that comes to mind is actually the opposite of your initial suggestion #2. Rather than moving the strings, what if you could "move" the entire bubble of normal space-time?
This would be to cause the whole pocket universe of your star system to slide through webspace, changing the location of the wormhole endpoints in your system, then causing them to be gone entirely, then maybe (depending on the density of strings in webspace) seeing new wormholes appear as you move your star system onto a new string.
If you had a lot of free time, eventually this would allow you to map webspace. That is, you could create a version of a 3D diagram of the local strings by observing which wormholes appear as you move through webspace and testing where they lead. I say a "version" of a map because, due to invariant-time travel, you couldn't judge distances; you could only know the general conformation of strings in webspace -- tentative maps showing the general location of strings.
That could be pretty useful.