Full model name of your motherboard? Your CPU+RAM is more than enough for any existing and upcoming in next two years games
Personally i don't see any reasons for system upgrade (except GPU)
1.Uninstall any software for GPU and system tweaking
2.Delete C:\AMD folder
4.Run HWinfo64>sensors only>Logging start and play for 15-20 min. I want to see that log
Can you run the game windowed? Can you run the game in borderless fullscreen? Do you have v-sync enabled in drivers or game settings? Do you have radeon chill enabled or disabled? Are you sure you run the game on radeon graphics and not on intel integrated GPU?
I used to have this GPU and I can assure you that LoL runs at like 300fps on it, depending on settings and resolution.
Also does this happen with other games, too?