Metro does run a little smoother with nVidia, but your specs should still be sufficient to run at High @ 1080p.
How do your temps look? (CPU & GPU - are you monitoring? If not, use something like HWMonitor and ensure they're in 'safe' rage).
As for Doom, your CPU is definitely holding you back. Minimum spec (from an AMD standpoint) is an FX-8320 or 8350.
Your GPU is still totally capable for what you want to do (at 1080p).
Ensure you have the latest drivers installed, including uninstalling old versions (you can use a program like DDU to do this).
Moving to 1440p, assuming playing the same games, you'll need to better your CPU - An i5-6500 is a great choice.
The 280X has some limitations at 1440p - you won't be maxing out new titles & achieving 60FPS, for example.
Should be more than a playable experience all-round.
I'd look to replace the CPU before the GPU.
As an FYI, as much as I hate to say it, an upgrade within the FX line - would likely be more 'budget friendly' - although not a very good 'bang for buck' all things considered.
An i5-6500, new motherboard, DDR4 RAM and likely a new copy of Windows will set you back between $350-$400.
While that would be my recommendation for you, if the budget is tight, you'd see improvements by upgrading your CPU to something like an FX-8350 for ~$200.
While I hate recommending AMD as there's nothing worth investing in (at the moment, new Zen releases should give some decent alternatives to Intel) - for the 'budget' option, as you already have the mobo/RAM etc, it might just make sense.