Hi there gshutch,
you shouldn't swap your prozessor out just yet. The 5600G in your system is fine for almost any game out today if its not running to hot in your HP prebuild. The limiting factor will be your graphics card, which isn't able to hold up to your CPUs performance in games.
On another note, it seems possible to me that the standard 310W powersupply in your HP prebuild could be the cause for your crashes. 65W CPU, 130W GPU and about 100W for everything else doesnt leave you with alot of headroom. This also limits the upgrades possible in your current PC alot, because your motherboard uses an unusual power connector, that standard powersupplies don't have, and most current low/midrange use a 8-Pin power-connector that would use too much power for your system.
My recommendation is to keep the system around for a while, and build a new PC with your old 5600G, RAM and SSD. So get a new PC-Case, CPU-Cooler (a 30$ one should be more than fine), AM4 Motherboard, 500-600W Power-Supply and either a 6600xt or 7600xt.
Hope this was helpful, and not too devastating.
