Here's an image that describes what I mean. The cap of the line segment sticks out when the line segment ends right next to a different line, and I was wondering if there's a good way to make it flush using Inkscape.
Please correct me if I'm wrong, but it looks like you're using the 'Triangle in/out' shape setting from the toolbar of the drawing tools.
Inkscape automatically adds the 'Power Stroke' path-effect to your path to achieve this style. That means that, while in node-editing mode, you can simply move the pink diamond-shaped handle that controls the thickness of the 'Power Stroke' effect a little backwards so it tapers off at the end.
If you don't like that line-cap style, then you can open the 'Path → Path Effect...' panel and set the 'End cap' (or 'Start cap') to 'Round' for example.
By the way, you can Ctrl-click, then drag these nodes to add more of them and fine-tune the thickness of the stroke along the path. Clicking them while holding Shift opens a little dialog where you can set the position and thickness precisely and a Alt + Ctrl-click deletes them again.
Admittedly this is only to make it less visually obvious. To truly get what you want, you'll probably have to use 'Path → Object to Path' (or 'Stroke to Path') and manually edit the nodes at the ends. Or cut those parts off with some helper objects and using 'Path → Difference'.