Do 'already' and 'just' require the present perfect?

Firstly, get the tense right for the purpose, then add just, already, yet, not yet, still, no longer, any more as appropriate.

If your purpose is to give news to someone, you will probably use present perfect in UK, past simple in US.

There is often a choice. E.g. we (have) already stopped sending her flowers = we no longer send her flowers = we don't send her flowers any more.

The use of the past tense in this construction, especially when spoken, is often used to express emphasis and, perhaps frustration.

We already GAVE him a response.

We just GAVE him a response.

These forms are used to indicate that the recipient is somehow not being reasonable in continuing to seek further feedback. The sentences would have less "punch" if they were delivered as

We HAVE ALREADY GIVEN him a response.

We HAVE JUST GIVEN him a response.