It's not true. You can easily check it in GAP, for example.  Here's a way to get a counterexample in $\mathbb{F}_9$ for $n=4$.

`FF:=GF(9);`
`sets:=Tuples(FF,4);;`
`sets:=Set(List(sets,Set));;`

`sets:=Filtered(sets,x->Length(x)=4);;`

`PowerSums := function(vals,n)`

`return(List([1..n],k->Sum(List(vals,t->t^k))));`

`end;`

` S:=First(sets,x->ForAny(sets,y->(not x=y) and PowerSums(x,4)=PowerSums(y,4)));`

This returns a four element set, and we can find the other one similarly.

`T:=First(sets,y->(not y=S) and PowerSums(S,4)=PowerSums(y,4));`