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 $n=4$.
q:=9; n:=4;
FF:=GF(q);
sets:=Tuples(FF,n);;
sets:=Set(List(sets,Set));;
sets:=Filtered(sets,x->Length(x)=n);;
PowerSums := function(vals,m)
return(List([1..m],k->Sum(List(vals,t->t^k))));
end;
S:=First(sets,x->ForAny(sets,y->(not x=y) and PowerSums(x,n)=PowerSums(y,n)));
This returns a four element set, and we can find the other one similarly.
T:=First(sets,y->(not y=S) and PowerSums(S,n)=PowerSums(y,n));
We also get examples for $n=3$ by just changing that one value at the start and rerunning.