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));