You would appear to need to concentrate on the suspect cylinder 1st.  Then from your posting, the fact that you can tilt the rockers side to side is not good. At the least the shaft needs replacement, and probably some or all that sides Rockers. As noted, due to poor previous oiling possibly, you may have badly worn valve guides or a bad valve seal.. Valve guide replacements are available, but require machining to install. There are guide liners available, which might work, but our moderator Ted, recommends one type over the other, & I don't have that info at hand at the moment..Good luck.    


Paul

I think after you get it all back together you should put a vacuum gauge on the engine it will tell you tons more than a compression gauge will. The readings are true indications of what is wrong .

I remember seeing a service bulletin from Ford about unequal oiling from side to side.  As I remember, a certain combination of camshaft and cam bearing caused one side to over oil and the other to under oil.  Their fix was to put a restrictor in the oil passage in the rocker stand of the over oil side to try to equalize the flow side to side.

John - "The Hoosier Hurricane"

"...the only time I've seen smoke is when I put the front of the car in a steep downhill drive to turn around and the wind blew a puff by me. It doesn't smoke otherwise."

I think what you will find is that the oil supply is overpowering the cylinder head forward drain and is pooling on a downward slope so as to flood the #1 cylinder valve seal(s). As the vehicle levels out, the smoking subsides. 

The valve guide(s) and or seal(s) may or may not be bad.          

____________________________

I think you are right. Because of the condition of the rocker shaft on the right side I think the engine was not oiling correctly on that side, leading to excessive wear on the rockers and valve guides. That being said, the only time I've seen smoke is when I put the front of the car in a steep downhill drive to turn around and the wind blew a puff by me. It doesn't smoke otherwise. 

I may be wrong but you could still have decent compression in a cylinder and still have a wet plug if the compression rings are good and the oil control ring is cracked or broken also if oils is dripping down the valve guides your compression will be fine. I think your engine was rebuilt, the heads were removed and washed down in the parts washer the crud that builds up around the valve stems was washed away the heads were reassembled with new seals and no machine work was done . I think if you remove the keeper and the valve spring on the cylinder you have the fouling on you will find excessive play in between the head and the valve stem. The fact that you have the heavy smoke condition and the high oil consumption makes me think your issue is valves not rings.  But I am just a 70 year old guy so do not quote me .

1955 Ford Crown Victoria Glass Top
1956 Ford Club Sedan Daily Driver
1971 Volvo 1800 I have owned this car 40 years 
1966 BMW 2000cs  under restoration                                                                       I cant afford a new car 

I just re-read your original Post. If You have One only oiled Plug as You indicated that would be the Cylinder to check Compression on. If You have the type of Compression Gauge that screws into the Plug Hole testing is not that difficult. If only One Plug is oiled possibly there is a broken Oil Ring or One not properly installed with the Expander not keeping tension on the Rings. Or it might just be the Valve stem Seals not installed properly or split if they were not replaced. If you only getting Oil Smoke from One side that at least gives You an Indication of which side the problem is on.

  

No, I haven't done a compression check. I know I should have already but I don't relish it with the plugs so hard to reach. I'll do it while I'm dealing with the rocker shafts.

When I rebuilt my 292 I had the new Camshaft oil Groove machined to I think .019 which is the recommended Groove Depth. As a result I got good oil Flow to the Heads. Too much actually. I had to tap the oil Feed Hole in the Rocker Stand and install a restricter to cut down the flow to the Shafts. If You are getting too much oil to the One side you could do the same. A small  Pipe plug with a hole drilled will cut down the Flow. The Rocker Shaft Stand is easy to Tap. Soft Metal. Have You done a Compression Test to get an Idea of what kind of Shape the Engine is internally? If You have even One low Cylinder that could be where the Oil is being burned.

  

The tubes are pointing into the push rod hole. This engine has the little catch pans under the rocker stands. It is evident that most of the oil is draining into the center two pans and onto the heads. From the condition of the heads and rockers, I believe a half-a$$ed rebuild was done on the engine. It might not have even been bored, just bearings and rings installed. In the end, I'll probably have to go through it again. It does have good oil pressure though.