Do you know what was used to impregnate the posts? If it was a water based product such as tanilising (not sure if you have that in the US or wheather it's called something else but it leaves the timber green in colour) then I would agree to leave it as the first reply said, an oil based product such as vac vac would be more tolarant.
What ever you cover the posts in I would advise treating it first.I wouldn't myself use ply for the covering, the edges would be a problem and not be very attractive even if rubbed down, filled and painted. Solid wood would be better and don't try to get the joints flush, you'll never disguise it. Take the front and back pieces and cut them to the width of the post and the side pieces should be cut to the width of the post plus the thickness of the front and rear plants plus half an inch to allow them to overlap by 1/4 " at front and back. Simple really, following the old adge that if you can't hide a joint then make a feature of it.