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));

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.