So it's your background that you have a problem with?
Well counting your background as a separate sprite, it would need to be 1.5x original size
meaning 1536 x 960 px. That should make it clearer but that is a huge waste of memory.
You should look for other options like tiled background.
I don't use Photoshop but I use Inkscape and every time I re-export a PNG image it gets blurrier and blurrier.
Sorry but I'm not the right person to ask for Photoshop advice as I'm an amateur.