I'd try to get a 5800X cheap on an auction site, they are being sold for around £300 by some sellers, without a box at the moment (obviously, check seller feedback), a bit more new + boxed. I don't think going from 8>6 CPU cores would be an ideal upgrade.
It's actually one of the best CPUs you can buy (but expensive).
You just need to buy a decent cooler with it, something like a NH-U14S air cooler is recommended to keep the CPU under 80 degrees at full load, but they don't come cheap. You can buy them preowned for ~£40 on auction websites.
It would be a good idea to update your BIOS before installing the new CPU.