From what I've understood, when I talk about things that are not entirely mine I have to use the term 우리. In formal situations in which I'd have to use the ㅂ니다/습니다 form do I have to use 저희 instead?
'우리' and '저희' means totally same. But the difference is coming from whether other party is higher than our.
If you're saying to senior/older people or formal situation you should use '저희'. ex) 저희는 한국사람입니다. - We are Korean.
You are same qualification(like friends) or older than other party, you should use '우리'. ex) 우리(나라)는 미국과 동맹국가입니다. - Our Country is an ally with the United States.
This usage is one of the many things that is wrong in Korea. When you confuse in grammar, You can simply use '저희' in many situations.