My 370 is collecting dust since I bought a 1080 TI.
I generally owned AMD GPU's, from 5770, 7870, 370, and finally the RX480.
I always did the DDU clean up and install, I wonder if that is what has kept me problem free.
I recommend you download the current drivers, perform the DDU driver cleanup, reboot into safemode, and use a program such as ccleaner to clean the registry of any old entries, delete any AMD folders that may still be present, then reboot again, back into safemode, where you install the new drivers.
Upon first boot up with the new drivers, let the system idle at teh desktop for several minutes, then simply perform a normal restart.
Go from there. Please post results.