If a wormhole could be described as two mouths connected by a tube of some length shorter than the mouths are separated in normal space, how is the "tube" able to be shorter than the normal space distance between the mouths? Is it travelling in a "straighter line" while normal space is curved, like cutting through a hill, or tunneling through a sphere, or are there extra spacial dimensions that connect them but disappear on the macro-scale?
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or are there extra spacial dimensions that connect them but disappear on the macro-scale?
No. Extra dimensions are not a feature of general relativity. Sometimes people draw embedding diagrams or conceptual diagrams that include extra dimensions, but this is just a visual device.
Is it travelling in a "straighter line" while normal space is curved, like cutting through a hill,
The surrounding spacetime can be flat (or, technically, asymptotically flat). The fact that a surface is represented as curved on an embedding diagram does not necessarily relate at all to whether it's actually curved. There is a distinction between instrinsic curvature and extrinsic curvature. Only intrinsic curvature is of interest in GR.
Distances in GR do not behave the way they do in flat spacetime. Embedding diagrams of wormholes really just show the topology, which is separate from the question of distance measurements. You can take the wormhole topology and outfit it with a variety of different metrics. The metric is the mathematical machinery that defines distances. If you want to, you can define the metric in such a way that the distance through the wormhole is longer than the distance through the ambient space. (And by default, a wormhole would probably transport you through time as well.)