The HD 6450 is too weak to play WoW and pretty much all games except older titles with any decent detail levels, and Windows 10 does not support cards previous to the HD 7000 series.
As far as your PSU goes, it is inadequate for the 370. Assuming it is able to give 460w, which it isn't, at 12v, which it can't, then given 4 years of age which cuts ~2% max rated power per year, you're only looking at 425w. Get a Corsair 650w PSU and it'll work.
Here are the legacy drivers that you can use for a HD 6xxx card:
Those are for WIndows 10 X64 which I assume you have since you have 8GB of RAM
Reducing the number of monitors wouldn't boost your gaming performance by the way since it doesn't take much graphics performance to display something on the desktop on the 2nd monitor.
You didn't say what resolution you would be using so I can't really say if it would play World of Warcraft or not. The new game expansion coming out the end of August improves the graphics for the game a bit so it is possible it may struggle then since the draw distance will be increasing.
It could be the PSU but firstly make sure your motherboard BIOS is up to date and if not update it as some BIOS updates resolve GPU incompatibility. You didn't say what motherboard you have. If it were me I would google that motherboard against the R7 370 and see if others are having the same issue. I would also be testing the R7 370 card on a friends machine to make sure it works or not. Depending if it works in another machine or not you can then progress further with replacing the power supply.
The old and new GPU can't share drivers so you should only have the new one in once you have it working.