Any tips on getting fresh drivers for my hd 7950? Currently using a fresh install of Ubuntu 16.04.
Are you having any specific problems? Normally the default AMD driver should be good to go.
If not: AMD Linux Driver Download
I'd like to do some gaming, and the latest driver for my card is for ubuntu 15.04 from 2015.
http://support.amd.com/en-us/download/linux
Am I just screwed until an update?
Hm, my guess is there will be no update, but did you try that driver? Chances are it still works. There should be one in the repository too.
Isn't that a GCN 1 card? AMDGPU has experimental support for GCN 1, but you need to compile the kernel yourself.
Edit: So one day AMDGPU (which is the default) will support your older card and you don't need any driver installation. My guess is Ubuntu 20.04. 
I recommend giving the latest Mesa drivers a shot: they have been truly tremendous as of late.
Oibaf has a well respected PPA for Ubuntu: https://launchpad.net/~oibaf/+archive/ubuntu/graphics-drivers