If the APU Driver is compatible with your GPU Card then you all you need to do is install one AMD Driver for both.
The last supported APU driver for your A4-6300 with HD8370D is from 2015 and 2016: https://www.amd.com/en/support/apu/amd-series-processors/amd-a4-series-apu-for-desktops/a4-6300-rade...
This is for the AMD Driver from 2016 BETA:
Seems like the R7-350 is a rebrand of a R7-250 . The compatibility list doesn't show any R7 200 or 300 GPU cards.
Also I found out that the R7 250X is a rebrand of the HD7770 GPU card which shows not to be compatible in the image above.
But your R7-350 has much newer last supported AMD driver from 06/21/2021: https://www.amd.com/en/support/graphics/amd-radeon-r7-series/amd-radeon-r7-300-series/amd-radeon-r7-...
But, I don't believe that the R7 350 last supported driver is compatible with your APU HD8370D Integrated Graphics.
So you would need to disable your Integrated Graphics in BIOS Settings and just use your R7 350 GPU card with the 06/2021 Graphics Driver.
You can't install 2 incompatible AMD drivers on the same PC. There will be a conflict when trying to install both on the same computer.
There might be a method to do it which I have read for laptops with the same situation as you. You can try and google it or try to find the threads here at AMD Forums.
Best is to just disable your APU integrated Graphics and install the newer 06/2021 AMD Graphics. If the game or games requires newer driver than 06/21/2021 then you will need to upgrade your GPU card to one that is still supported by AMD.