The normal way to think about cosmological redshift is as an expansion effect. It is handled pretty much as your post suggests.
During transit the light's wavelength increases by a factor of 1+z, and that happens if distances in the U have increased by a factor of 1+z while the light was traveling.
Cosmologists use a time dependent number a(t) called "scale factor" to keep track of distances.
If z = 1 that means distances (and wavelengths) have doubled. That means the ratio of scalefactor NOW a(now) to scalefactor THEN when light was emitted is
a(now)/a(then) = 2. Distances now are twice as big as when light was emitted and started on its journey to us.
These are LARGESCALE distances between widelyseparated points each of which is at rest relative to the cosmic background radiation. Not smallscale distances within our solarsystem or galaxy.