An American would say almost any of the following:

I'm lucky.

I'm so lucky.

I feel so lucky.

I have good luck.

What good luck!

Someone who is more religious would say something like "I'm blessed" or "I'm so blessed".

Normally a person doesn't say "I'm a lucky dog" about themselves.  However, they will occasionally use it to refer to someone else, often man to man: "his girlfriend is so beautiful, what a lucky dog!" or "congratulations on getting a raise; you're a lucky dog!"

Thanks so much for everyone.

I have learnt what I want to know.

Thanks for your help.

Do you have more context to refer to?