I apologize that I SHOT your dachshund.
If you say enough apologies, I won't shoot you.
|( に ) あやま*る||
apologize (BOOBOO: This is only used when describing actions after the fact. You wouldn't step on someone's toes and then say わたしはあやまります！)
|に 感謝 する||
The most common, general word for gratitude.