Planar Network Download

New to OSM usage, so kind of a newb. Can someone provide me with steps to download a single US state worth of vector data so I can build a planar network from the OSM data. I’ve downloaded a subset of data using a rectangle selection, but the resulting .osm data when viewed and queried in QGIS is not a complete planar graph. For instance, when I try to select a single vertex the resulting line transcends across multiple vertices. Is OSM data a true planar graph or will I have to run an intersect algorithm on it? I’ve tried the QGIS intersect function on the sample set I downloaded, but it crashes. My end goal is to have a bounded administration vector layer(state level in my case) with all vectors terminated at all intersections and no dangles outside of the bounding admin boundary.

Attached picture is what happens when I try to select a single feature in QGIS from the downloaded sample I got from OSM. The selected vector as you can see spans across multiple vertices without any breaks.

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