Bounding the exponent of finite $p$-groups with normalizer conditions on cyclic subgroups

Question

Suppose $P$ is a non-cyclic finite $p$-group satisfying the following two conditions:

1. All cyclic subgroups of order $p$ in $P$ are normal (this is equivalent to saying that $\Omega(P) \subset Z(P)$).
2. Any cyclic subgroup $C$ of order $p^2$ has exactly $p$ conjugates, i.e. $[P:N_P(C)] = p$.

The second condition implies for example that $\Omega(P) = Z(P)$. I would like to know whether these two conditions are sufficient to bound the exponent of $P$. Using GAP I could not find a group $P$ satisfying the above conditions but having exponent larger than $p^2$. If you like, you can also assume that $p \neq 2$ if this makes it easier.