Grammaticality – Which is Correct, With ‘Have’ or Without ‘Have’?

Your question concerns ellipsis, omitting one or more words obviously understood but needed to parse the grammar. For this to occur, you need two strictly parallel elements joined by a coordinating conjunction, including than.

You are asking how much you can omit in the sentence

I have been admitted to the upcoming class and I have been planning to live in the dormitory.

Omitting the I in the second clause isn't really ellipsis. All you're doing is using a compound verb with a single subject.

Whether you can omit have and/or been from the second verb phrase depends on whether have been admitted and have been planning are in strict parallel. But they are not: the first is present perfect passive voice and the second is present perfect progressive, active voice.

That leaves you:

I have been admitted to the upcoming class and have been planning to live in the dormitory.

I suppose you have your reasons for not writing

I have been admitted to the upcoming class and plan/am planning to live in the dormitory.

This would be the most idiomatic way of saying what you want to say.