Haha yeah, "AMD RS780" got me confused for a moment since there are AMD RX480, 580, etc GPUs... The ATI (before the rebranding in 2010) RS780 from 2008 is indeed a bit out of date 
Are you sure that's the discrete GPU though? RS780 seems to be an onboard GPU. Could it be that the dGPU is not detected because of a missing driver or something? In doubt check inside the case
For hashcat you'll most likely need a new PC as I am not sure a new GPU would work very well with a 2008 CPU and PCIe 2.0 motherboard.
32GB of RAM is a very nice amount, so don't throw away your PC: you could use it for something else like a NAS or maybe a dedicated streaming PC. Don't make the mistake of trying to use these 32GB in a new PC: it's DDR3 while new PCs use DDR4.
Are you sure that's the discrete GPU though? RS780 seems to be an onboard GPU. Could it be that the dGPU is not detected because of a missing driver or something? In doubt check inside the case
For hashcat you'll most likely need a new PC as I am not sure a new GPU would work very well with a 2008 CPU and PCIe 2.0 motherboard.
32GB of RAM is a very nice amount, so don't throw away your PC: you could use it for something else like a NAS or maybe a dedicated streaming PC. Don't make the mistake of trying to use these 32GB in a new PC: it's DDR3 while new PCs use DDR4.