Aaaaargh rook endings!

I just lost this rook ending (time control was 15|10) in an intensely frustrating manner. I went for the rook ending at around move 27. I was two pawns up and had a passed pawn on the a-file, and surely at worst I should have come away with a draw. Instead I managed to lose. Any thoughts on what the right technique would have been would be very welcome, as well as any thoughts on which moves of mine were particularly bad. I've included my initial thoughts in the analysis.