Any recommendation depends on your location - I wouldn't provide the same answer for Yuma, AZ as I might for Edmonton, AB.
When water vapor diffuses through a wall and reaches the exterior sheathing, if that sheathing temperature is below the dew point the vapor will condense. Depending on the amount of water vapor and the rate at which the wall cavity is able to dry out, this may lead to moisture problems.
Installing insulation in the wall cavity will help reduce heating costs, but won't address the problem of sheathing temperature. As a result, some builders are specifying 2" to 4" or more foil faced polyiso panels be installed outside of the sheathing to raise the temperature above the dew point. In these designs, some don't even advocate using cavity insulation. Vapor will still diffuse into the cavity, but it will not condense on the sheathing surface.
Installing foil faced polyiso on the interior of the wall, properly air sealed and with taped joints, forms an effective vapor barrier which will reduce the problem of moisture diffusing through the walls. However it may not solve all of the problems with vapor diffusion so it's quite possible to still have problems.
Additionally, while most people look to insulation as a primary solution for energy savings, often older homes would find greater benefit from air sealing. There is almost no way you can make an older house with plaster walls "too tight." In fact, the idea of houses being "too tight" is something of a myth. Your goal should be to have "managed ventilation" rather than the unmanaged ventilation common in older homes.
In my 1870's New England house as an "inside" solution I removed the plaster, installed fiberglass batt insulation in the cavities, applied 2" foil faced polyiso over the studs (for a thermal break and vapor barrier), with 1x3 furring strips and gypsum on top.
I can't quite picture what you're describing, but covering studs is always better than placing insulation in between when doing so is practical. Studs do not provide good insulation and turn into a heat sink. The dead air space left also provides a bit more insulation. Even better would be to insulate that space, then put more insulating sheathing over that and the studs.
Best Answer
This sounds like a good candidate for the procedure you outline, done with loose fill cellulose insulation, with two caveats.
First of all, while this could conceivably be a DIY project, you can't rent a typical cellulose blower from Home Depot or the like because it doesn't have the power to really densely pack the stuff in there; as a result, the material will settle and you'll wind up with an uninsulated band on top. You need one of the more powerful professional machines that can do wet-blowing and dense-packing.
Secondly, you live in a rainy climate, and it's likely that the empty wall cavities have been permitting the boards of your wood lap siding to dry readily when they get wet due to convection within the wall and heat leaking out from the interior. These traits are bad from the perspective thermal comfort and utility bills, but fixing them will in all likelihood reduce your walls' drying potential, which could lead to your siding rotting and the paint peeling. If your house has very wide roof overhangs or a wraparound porch or something such that the walls rarely get very wet, this may be no big deal. But if not, the outlines project is much riskier and it may be worth it to expand the project and remove the siding so you can install modern sheathing with a weather-resistant barrier like double-layer tar paper or Tyvek. And if you're going to do that, you might as well add even more insulation in the form of several inches of rigid foam or mineral wool boards over the sheathing--or even in lieu of wood sheathing, in the case of foam!