At what stage should mushroom logs be soaked?

The dry season's coming up here- 5 months of hot, dry rainless days. I don't want to wait another year to inoculate logs with mushroom plugs though.

Should I soak the logs before putting in the dowels, right after they'r in, or months down the track? I'll keep then in full shade but they'll still dry out.

There are two different stages to growing mushrooms.  If the logs are freshly cut, you don't need to soak them.  There will be enough residual moisture in them to keep the fungi happy and start a fungal network within the wood.  If the logs are old and dried out, it may be too late, as other fungi may have already colonized them through cracks and checks in the wood.  Spores are everywhere: they will have most likely have landed all over your logs and found their way into any little nooks and crannies.  If you soak them to rehydrate them, you may be feeding a fungal colony not of your choosing.

However, if you start with fresh logs and plugs, once inoculated and sealed, you need to keep the logs moist.  Some people who live in dry areas (like what you describe) use a mister that comes on regularly with a timer.  They stack the logs in a log-cabin/hashtag # pattern in a place where they'll be in the shade all day, and the mister comes on several times a day.  After a year, if conditions have been sufficiently moist, the logs will be sufficiently infused with a fungal network.  You don't soak the logs, but just mist them regularly to keep moisture levels consistent.

After a year, to get a flush of mushrooms, you completely submerge the logs in a tank of chlorine-free water for a day or two.  Then you return the logs to the shade and continue to keep them misted as the fruiting bodies (mushrooms) pop out.  I've never heard of people soaking logs at any other point—just when they are attempting to get a flush of 'shrooms.  

Yup! they need to be moist otherwise the fungus won't grow. To keep moisture place them in the shade and bury them into moist send -when your logs are fully colonized (7-12 months depending on several factors like strain, climate, wood type) give em a cold shock -keep em for couple of days under water after that they should fruit.

Logs that your going to put plugs in should be cut and set up for about two weeks (probably one week in your part of mother earth) then drill plug and seal with wax (don't forget the ends).
Stack and mist often enough that they don't dry out, keep in as deep a shade as possible.
When you see white mycelium covering the ends of the logs under the wax, you are ready to fruit by soaking for 48 hours then standing the logs up at an angle (don't let them touch the ground, ever).
Be sure you know the natural fruiting times of year for your species, no sense trying to fruit if they aren't going to do so in nature, doing that requires rooms where you can control light, temp and humidity all year round.

Redhawk