Re "The Old Soldier Who Had a Wooden Leg", my mother sang a song that sounds like that, but the tune was "Turkey in the Straw". Her title was "The Old Geezer". The lyrics were:

There was an old geezer and he had a wooden leg,
And he had no tobacco, but tobacco would he beg;
Another old geezer was as cunning as a fox,
And he always had tobacco in his old tobacco box;

Said the first old geezer, "Won't you give me a little chew?"
Said the second old geezer, "I'll be darned if I do!
Just save up your money, and hide it in the rocks,
And you'll always have tobacco in your old tobacco box."

Those lyrics would likely also work if they were sung to the "Red Haired Boy" tune.