I've been told that xem can be used in place of thấy - to see and also nhìn - to look at. Is this true?
That is kinda true. Xem is very commonly used in North Vietnamese instead of thay or nhin but in general (in the case of seeing) thay is referring to suddenly seeing something and indicates a very short timed action while nhin and xem would last longer.
So..... I can't use xem for "You watch (look after) my shoes while I go swimming"?