The AMDGPU driver no longer supports the TeraScale 2 architecture used on the Radeon 7310.
Neither of the above support Vulkan.
Vulkan requires GCN (Graphics Core Next) 1.0 or newer, which replaced TeraScale.
You also cannot easily toggle between older and newer drivers. So while you can enable the Radeon driver, it will mostly only work with older games.
You must add a kernel parameter, since the AMD drivers are included in the Linux kernel.
sudo nano /etc/default/grub
Arrow key down to GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" and move the cursor to the end of splash and tap space once to space over. Now paste in
radeon.si_support=1 radeon.cik_support=1 amdgpu.si_support=0 amdgpu.cik_support=0
That parameter sets the Radeon driver and disables the GCN AMDGPU.
Tap ctl+o to overwrite, then enter key to save current configuration, then ctl+x to exit the editor.
Now you must run
sudo update-grub
for the changes to take effect.
Remember: You will be using OpenGPL, Mesa, not Vulkan.