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!"
Student or Learner