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.