I did use a few continues, so maybe I have to beat it on one continue to get that ending. I did some searching on the internet, but I couldn't find anything. Anybody know the answer?
I'm very sorry I don't know the answer to your question, kind sir.
I absolve you.
I pulled the game out again and beat it using one continue, and using zero continues, and you do get the special ending only when you don't use any continues.
If you use one or more continues, Nick and Tom rescue the princesses and get friendly smiles from them, and they banish King Scorch and return home.
If you don't use any continues, the princess kiss Nick and Tom after they rescue them, and the kisses melt the snow and turn them back into their original form. King Scorch is again banished, but this time Nick and Tom marry the princesses and live happily ever after. Or, just look at the ending linked in the original post — it's the same one.
Well, glad to have that out of a way — a pretty fun game, overall (even though the colors are hideous).