I believe, 'it's me' is correct in modern usage. 'It is I' is a weird thing you might see in a historical drama. It's not necessarily wrong, but it's very formal, and very old fashioned.

There are examples here where you might use 'It is I' as part of a longer sentence - see subtitle 'Examples of It is I in a sentence'. https://www.thesaurus.com/e/grammar/it-is-i-vs-its-me/

In these cases, you'd take out the 'who' in your mind, so it would read 'I should...' because that would make sense where 'me should' does not.