Those 2011 MBA have pretty bad boards, giving all sorts of issues with graphics and RAM. Not worth fixing and in my opinion not a good idea to buy a replacement board either. Unfortunately the 2012 model which runs fine has different screen cables. If you can find a used late 2010/early 2011 machine with C2D board 2.13Ghz and broken screen you can swap yours, that’s where I would put some money to keep part of the value. Or sell your screen assembly and topcase and buy a later one.