The mean exactly the same thing and are both correct. "I don't have a plan," is more common, at least in the US.

Both are correct. I don’t have is more common in the U.S.

I haven't a plan doesn't exist. I have no plan

"She has been eating noodles". In this sentence, have is an auxiliary verb. We use "have" as an auxiliary verb in perfect tense forms. "She has a daughter." In this sentence, have is the main verb. When "have" is the main verb, you can use "do" with the negative form. "She doesn't have a daughter." It's not incorrect to use "haven't", either. "I don't have any money" "I haven't any money."