need help with endgame to avoid draw

Or 41.Ra5#

41) Ra5# in the first game

instead of 21...b6 get the other rook out into action Rd8

You're dropping too many pawns. Trade rooks even if it costs you a pawn

53....e3 ?? better was 53...Kd3  54.Kf7 Rb8  and you king can collect the f and g pawns and your rook will stop the promotion on the g8 sq.

Animal kingdom was black, but yeah I can't believe his opponent missed the mate!!

Let me try some variations:

In rook and pawn endings like this the rook activity is very important and can be worth several pawns in some cases. Priority one is to restrict his rooks first and then aim for activity for your rooks then trade the rooks for a winning king and pawn endgame.

35.Rb8

to execute as devoid pointed out that 24....Rd3 is far better than Rd8 and c5

I don't know kco, by how much Rd3 is better. Both black's rooks are pretty inactive, and they both need to be activated. I suppose Rd3 avoids a rook trade so it is better...

This is true. Black certainly cannot win without the e-pawn as he has to eventually sac his rook to passed g-pawn. Line 53... Kd3 54. Kf7 Rb8! 55. g7 Ke2 results to easily winning pawn ending for black. However, white has a more cunning defence in 54. Kf5!? since it gives black possibility for mistake with 54... Ke2? when 55. Kxe4 is actually a draw (for example 55... Rxg6 56. f4 Rxg3 57. f5 etc.). Black should still win with accurate 54... Re8!

Because of the above winning idea 51... Kd3 would look more logical than 51...Kd4. Around move 24 the method suggested by devoid looks most technically correct for me too.