You only have overlap of valance and conduction bands in metals (e.g. copper) and marginally so in semi-metals (e.g. graphene, tin). That's why they are "conducting" - there are lots of free electrons floating around from the valance band that happens to already overlap as an "electron gas" around the lattice ions. This is NOT how semiconductors or insulators conduct carriers however.
Stop thinking in terms of shells; they are a thing in chemistry but they will only be confusing because "shells" sound like physical orbits or distances in classical mechanics, and they aren't that. A carrier to change in energy without even changing location.
Instead the key is carrier energy and what's allowed by the lattice they occupy. This is where stuff like Brillouin zones and crystal structures enters the picture. The regularity of crystals constrains carrier movement something like how your line of sight through an orchard depends on the particular angle at which you look through it: some angles are obscured and others you can see clear to the other side. The regular lattice of a crystal conspires with the wave-nature of electrons to cancel out certain directions which is the same as forbidding any motion which results in band gaps at certain energy levels (which by quantum mechanics relates to electron "wavelengths" by hc/lambda).
Another way to think of why band gaps exists because quantum mechanics forbids certain energy levels from being occupied. If you look at electron spectral lines of a single atom (e.g. the classic hydrogen lines) you get characteristic lines corresponding to particular energy. You can never get energy levels between the lines.
However, bands exist in solids because you can't have the same energy/location occupied at the same time (because electrons are Fermions and follow Pauli Exclusion) so two electrons in an otherwise identical circumstance (i.e. in a crystal lattice) cause the one "spectral line energy state" to split in two, with one energy level above and one below the original line energy. Add two more and you get a 4-way split. Continue this to an Avogadro Constant number of atoms and electrons and you get energy bands. So these energy bands simple "are" - don't try to think of them beyond the "quasi-quantum" explanation. Quantum mechanics pervades semiconductors utterly so classical mechanics pretty much always fails as a model or analogy.
In semiconductors, there's a band gap between the valance and conduction bands so you ONLY get conduction if the carriers have enough energy to "cross" the gap in a single instant (no partial crossings allowed). Note this "gap" is not a physical gap but an energy gap so the carrier can be "unmoving" and magically get some energy from somewhere and then suddenly it is in the conduction band and on its way, susceptible to forces that had no effect on it before. The magic energy source, for instance, can be a photon interaction (e.g. a photovoltaic cell), statistical thermal agitation by not being at 0K temperature or merely a sufficiently strong electric field. The difference between valance or conductance occupancy and reactivity is almost like the carrier didn't exist in the valance band but was "summoned" into existence in the conduction band!
Doping actually drops little archipelagoes of energy levels into the gap which facilitates conduction and provides "cheater" carriers already near the conduction band that only have to hop a small energy distance to become conducting. Typically thermalization (being at 300K) does the trick of completely ionizing the dopant atoms and freeing the carriers into the conduction band. The term "freeze out" is when this thermalization reverse with (usually) cryogenic temperatures and the semiconductor becomes a "stone cold insulator" because none of these carriers can thermalize.
BTW if you make the gap bigger (because the atoms involves dictate it by "dumb luck of the periodic table"), there is nothing that actually differentiates a semiconductor from an insulator other than the band gap size and what happens to be relevant electrically. So for example, SiC (silicon carbide) can be deemed an insulator or a semiconductor depending. In the former it makes a decent insulator. In the latter you can make blue LEDs and kick-ass power MOSFETs. Similarly diamond can be used as either an insulator (used in some silicon processes) or a semiconductor (at high temperature once it have thermalization of carriers to allow traversal of the band gap). Just a matter of degree.
One final thing: once you have a carrier (electron or hole) in the conduction band, the primary mode of movement is a combination of classic diffusion with a little bit of field-induced drift. So think of a drop of magnetic ink in a glass of water where you have a magnet on the side of the glass- that's how carriers move in solids: metals, semiconductors and even insulators. Yes, all insulators conduct carriers, just not well and not without some destructive effects in doing so - this insulator "conduction damage" is related to what causes semiconductor devices to eventually fail - and because of quantum mechanics you can't prevent it!
So thinking in terms of a mechanical particle movement analogy will only get you into trouble - eventually your analogy will utterly fail and then you stop learning/understanding. Better to grok the deeper models a bit.