In both situations I would use the second sentence.
Which one shall we choose in each case:
(1) It's I who you are looking for.
(2) It's me who you are looking for.
(1) It's I who should say thank you.
(2) It's me who should say thank you.
In both situations I would use the second sentence.
Thanks for your prompt reply, florr0. I was wondering if we need to look at the normal word order. Normally we say You are looking for me. So we should choose (2) It's me who you are looking for. But for the second case, I guess we should choose (1) It's I who should say thank you, because normally we say I should say thank you. What do you think?