I would say either, but not both on the same shirt.
As time goes on if you are fortunate enough to get a national FA appointment (FA Youth Cup, Ladies FA Cup, FA Cup, FA Vase) then you are expected to wear the county badge.
The FA badge can be worn for other competitions.
I've been thinking while typing this, and would actually suggest the county badge for the reasons above, but also because IMO an embroidered county badge often looks better than a sew-on badge. It's certainly that way down here. If you get a second shirt, then either have that embroidered or use the sew-on FA badge which looks better than the county version in that form.