You only need to wear your county badge on county appointments or FA appointments. For league games there is no obligation to wear a specific badge and you could wear FA, county, or both (although I personally think two badges makes a referee look like a boy scout ..!). And some leagues have their own badges as you say.
Once at level 4 you are unlikely to wear county badge for league games as the FA require the FA badge for contributory league games. When I got FA Cup / Trophy / Vase appointments I would tape the county badge over the FA badge (and by the FA badge I mean the black and white one that is routinely embroidered on kits). The coloured three lions FA badge is only awarded to officials who have been given a major FA final.
