A: I don’t know how to thanks you. I owed you a lot.
ⓐ I will return your kindness one day.
ⓑ I will owe you one someday.
It's "to thank you" (not "to thanks you"); and "I owe you..." not "I owed you...."
If you wish to continue your thanks, you might say option ⓐ above, but not ⓑ, as you already owe them "one" now (so, don't use the future tense "I will owe...").
B: Please, don’t say that. I did nothing. If you keep saying,
ⓐ I feel shame myself.
ⓑ I’m shy myself
Here, I think you might simply stop after the "I did nothing" statement. If you wish to continue for emphasis, then the more natural thing would be to say, "You embarass me!"