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We consider the stochastic evolution of a (1 + 1)-dimensional polymer in the depinned regime. At equilibrium the system exhibits a double well structure: the polymer lies(essentially) either above or below the repulsive line. As a consequence one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However the energy barrier between these two phases is only logarithmic in the system size L and therefore the two relevant time scales are only polynomial in L with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. In particular the usual scenario in which the tunneling time coincides with the exponential of the energy barrier breaks down. Our main results are: (i) a proof that the mixing time of the system lies between L^{5/2} and L^{5/2+2}; (ii) the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap.