I checked the fish profile for Bronze Cory's and it says: "Corydoras introduced to new aquaria will settle in better if the tank is established; Cory's do not adjust well to a new aquarium with still-unstable water conditions and fluctuations."
I also read on another site that pH changes of 0.3 over a period of 24 hours can be deadly to some fish.
Perhaps you can try to slowly adjust your pH in one or the other (maybe even both tanks) to be more or less the same and then add the fish?
Drift wood can lower pH, but you cannot predict how effective it will be and how quick it will work. It depends on the buffering capacity of your water and how many tannins your wood will leak into the water. I've had some luck with it, but my tap water has a lower pH than yours to begin with. I read that you can add peat moss to your filter media or into your substrate to lower pH. It sounds a bit messy to me, but maybe it works.
One (very) effective way of lowering pH is the addition of CO2. Since you have a planted tank, the addition of CO2 may be beneficial for your plants. Not sure if you have a CO2 system already or not. There are simple and rather inexpensive DIY systems made with yeast OR you can choose to invest a bit and get a CO2 tank.
With that said, if you decide to add CO2 make sure you can properly monitor how much is added. I just recently lost over 50 fish in a 55G tank
, because my CO2 tank leaked and (very effectively) lowered pH from a happy 6.8 to lower than 6 where my kit wasn't able to pick it up anymore. Surprisingly, the only survivor was a cory cat.
In the other tank you could raise your pH by adding crushed coral or shells or even small amounts of baking soda. Also, increasing aeration will drive out CO2, consequently raising your pH. Whatever you do, do it slowly.
As far as acclimating fish, below is a link to a useful article about "Properly acclimating fish".
I used drip acclimation on one of my fish recently and he responded very well. Properly acclimating fish
Good luck and keep us updated! Congrats on the new, bigger home for the fish!