gr.group theory - orders of products of permutations

Let $p$ be a prime, $ngg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set ${1,...,n}$ which together act transitively on ${1,2,...,n}$ and such that all products $w(a,b)=a^{k_1}b^{l_1}a^{k_2}...$ of length at most $n$ satisfy $w(a,b)^p=1$ (here $k_i,l_iin {mathbb Z}$)?

Update: Following the discussion below (especially questions of Sergey Ivanov, here is a group theory problem closely related to the one before.

Is there a torsion residually finite infinite finitely generated group $G$ such that $G/FC(G)$ is bounded torsion? Here $FC(G)$ is the FC-radical of $G$, that the (normal) subgroup of $G$ which is the union of all finite conjugacy classes of $G$.

For explanations of relevance of this question see below (keep in mind that the direct product of finite groups coincides with its FC-radical). Note that if we would ask $G$ to be bounded torsion itself, the question would be equivalent to the restricted Burnside problem and would have negative answer by Zelmanov.

If the answer to any of the two questions above is negative for some $p>665$, then there exists a non-residually finite hyperbolic group.