This is more as a tip than a solution.
If you are having trouble finishing a game of Mah-jong without shuffling, or if you notice you are down to 1 pair, and removing that pair won't open another match, then quit out of the game completely. Then, restart the game and do the game of Mah-jong over again.
This won't count as a fail and you can redo the same game of Mah-jong and try to complete it without shuffling.