When you moved to Windows 7 from Windows 10 was it a Clean install of Windows 7?
Here are the last 2 AMD Drivers for your GPU card: https://www.amd.com/en/support/graphics/amd-radeon-hd/amd-radeon-hd-6000-series/amd-radeon-hd-6450
Try using DDU (Display Driver uninstaller) in Safe mode (best method) or the Windows desktop with the internet disconnected and deleting C:\AMD installation folder if created.
Download the full AMD driver package and install it, still with the internet disconnected. Once it install successfully, reconnect the internet and again delete the installation folder C:\AMD.
See if the BSOD continue. If it does jot down the exact BSOD error code and driver if shown. Also check Device Manager and make sure you don't have any drivers with issues showing errors or yellow exclamation marks.
Are all your 3rd party Apps compatible with Windows 7 and it drivers.
As you probably are well aware of MS has stopped supporting Windows 7 OS. So eventually so will AMD. But since your GPU card are not supported anymore that isn't an issue unless you are planning on upgrading the GPU card.
With a monitoring program check the temperatures of the GPU card when it BSODs.
You might want to use OCCT to stress test your CPU, GPU, & PSU for stability issues or hardware defections. If it passes all 3 tests without crashing or overheating than most likely is an incompatible issue with Windows 7.