What’s the difference between “I don’t have a plan” and “I haven’t a plan”? Thank you.

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

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Both are correct.

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

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I haven't a plan doesn't exist.

I have no plan

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"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."