As already commented, you want something not possible. Every shape has it own nodes and stroke. In addition a path, where a single node has 3 or more branches, is nonexistent in Inkscape.
You must have open paths to avoid overlaps. Examples; parts are moved a little apart:
The black version has separate shapes. The green version is a single open path
You can draw everything easily with the pen if you have a grid for snapping or if you use point snaps and have a locked copy of your original attempt in the bottom.
If you want handle the black version as one, you can make a group or combine the paths. The latter is mostly used to make holes to closed shapes, but one can also make a combined path which keeps several non-overlapping paths together.