Show that a regular language L contains only palindromes if and only if all words of length at most 3n are palindromes
Suppose that $L$ satisfies your condition. We prove that $L$ accepts only palindromes by induction on the size of the word.

Let $a \in L$ be a word of length at least $3n$. Tracing some accepting computation of $a$ on the given NFA, some state must repeat at least four times, so we can write $a=xuvwy$ (where $u,v,w$ are non-empty), where the state after reading $x,u,v,w$ is the same, say $s$. So $u,v,w$ all have a computation that goes from $s$ to $s$. This implies that the NFA also accepts $xy,xuy,xvy,xwy,xuvy,xvwy$. By induction, all of these are palindromes, and so they satisfy the prerequisites of the lemma with $x',u',v',w',y'$ being obtained by reversing the unprimed string. The lemma then implies that $a$ itself is a palindrome.