These errors are usually raised when some DLLs are corrupted or missing or when they are replaced with an incompatible version. It may also happen that you placed a DLL file in the game folder that is loaded instead of the genuine one. Possible solutions:

Verify the integrity of the game cache using Steam.
Re-install DirectX.
Uninstall the Visual C++ 2013 Redistributable package (x86 version), restart your computer and install the package again (do not use the Repair function).
Run the Windows System File Check tool to repair corrupted system files.
If you are using post-processing tools like Reshade or SweetFX or utilities like RadeonPro remove them.
Re-install the game; remove all files from the game installation folder and then verify the integrity of the game cache using Steam.
To identify the precise DLL that is causing an issue, download a 32-bit version of the Microsoft Dependency Walker and open arma3.exe with it. The libraries with a red icon are the corrupted ones.

I ran Dependency Walker and it came up with multiple .dll errors. I've uploaded the report. I've re-installed, verified the cache and have removed sweetFX but that didn't work.

Hello,
please use Dependency Walker on Arma3.exe not Arma3Launcher.exe.

Oops, sorry, I ran it again on arma3.exe and there are still a bunch of .dll errors. I uploaded the report.

Hello,
the file "D:\steamlibrary\steamapps\common\arma 3\DXGI.DLL" is not part of Arma 3 installation. Please remove the file and run the game again.

Cool, that worked. Thanks.