How should I approach fixing this issue?
You have got trifold edges, that is edges that belong to three faces.
Not only is that a bad modelling practice, that creates non manifold meshes, that could never really exist in real life.
That causes severe modelling and topology issues and will lead to trouble sooner or later, least of which are the ones you are encountering with the Solidify modifier.
Eliminate the trifold edges, either by erasing the faces and recreating them by duplicating an edge; or using the Rip operator and separating the duplicate edges.
Also make sure your Solidify has the Even Thickness option ticked
Alternatively consider using bezier curve objects for this sort of work, they seem like a more adequate tool for this sort of structure.