A Dirichlet problem for a parabolic reaction-diffusion equation is considered on a segment. The highest derivative of the equation is multiplied by a parameter $\varepsilon$ taking arbitrary values in the half-interval (0,1]. For this problem we study classical difference approximations on sequentially locally refined (a priori or a posteriori) meshes. The correction of the grid solutions in the difference schemes is performed only on the subdomains subjected to refinement (the boundaries of these subdomains pass through the grid nodes); uniform meshes are used on the adaptation subdomains. As was shown, in this class of the finite diference schemes there exists no scheme that converges uniformly in the parameter $\varepsilon$ (or $\varepsilon$-uniformly). We construct special schemes, which allow us to obtain the approximations that converge " almost $\varepsilon$-uniformly", i.e., with an error weakly depending on $\varepsilon$:$|u(x,t)-z(x,t)\leq M[\varepsilon^{-2\nu}N_1^{-2+2\mu}+n_0^{-1}]$, $(x,t)\in\overline G_h$ , where $\nu$, $\mu$ are arbitrary numbers from (0,1]; $N_1+1$ and $N_0+1$ are the numbers of the mesh nodes in $x$ and $t$, $M=M(\nu,\mu)$.