Depending on how we use it. 我是累 can be valid in some context to clarify the fact that I am truly tired. E.g. 你是不是累了? 我是累(了).
Similarly, 我也是又累又渴 can also be valid in a context like A: 我是又累又渴 B: 我也是又累又渴.
Without previous context, we can say 我又累又渴. It's generally saying I'm tired and thirsty. It's not for clarification or anything.
So, this use of 是 is for clarification or emphasis. It would be a misuse of it when we don't need to clarify anything.
PS. As comments suggested below, in 我是又累又渴, 是 can be used either for emphasis or for a normal connecting word.