Basically the angle between the bottom and top wheels is too great causing the chain to skip to the outside of the bottom wheel under force. If I use the higher adjustment screw on the shimano derailleur to align them better the top wheel is then not aligned with the smallest sprocket.
I'm a beginner and had an attempt at working it out but probably missing something obvious. 
From the the photo of the derailleur it's obvious that the derailleur cage is not parallel to the sprockets and wheel. The bottom run of the chain returning to the crank is leaving the cage at angle.
The derailleur hanger that attaches the derailleur to the frame is bent and causing the misalignment, and I think the cage may be deformed as well.
There are special tools for realigning the derailleur that a bike shop can use, but it's possible to improve it by hand. From the back of the bike look down the chain. The cage should look parallel to the chainrings. You can grab the derailleur and bend it back into alignment.
The first thing to be checked is the derailleur's hangar alignment. A bent or misaligned hangar can produce this effect on the derailleur jockey wheels. There is a special tool to check and correct this, though one can fashion a homemade device to do the trick.
Park Tool's advice on their derailleur hangar alignment tool.
You intimated that you're using the limit screws to attempt to align the jockey wheels. This is only helpful on the two extremes of the cassette. The H limit screw should be set with the chain on the large chainwheel and smallest, outermost sprocket. The position of the upper jockey wheel of the rear derailleur should be aligned with the outside, vertical plane of the small cog. Loosening the inner cable fixing bolt so there is no cable tension involved prior to setting the high side jockey wheel position is necessary for the proper setting. Park Tool's site has a video on setting up a rear derailleur that further explains the process.
When the limit screws are set, adjusting the line up of jockey wheel and cassette cogs is accomplished by adjusting the Barrell adjuster, where the cable enters the derailleur. Another Barrell adjuster is often found at the shifter, again where the cable enters. Typical use of this one is when you're on the move, fine tuning to get smoother, quicker shifts or eliminate chain noise.
Another thing that needs to be determined at first is if there is a problem with the derailleur structure such as a bent cage ( the frame the pulley wheels sit in) or extremely loose linkages of the derailleur body.
