The leading term of asymptotics of prices and sensitivities of barrier options and first touch digitals near the barrier for wide classes of L\'evy processes with exponential jump densities, including Variance Gamma model, KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In particular, it is proved that option's delta is unbounded for processes of infinite variation, and for processes of finite variation and infinite intensity, with zero drift and drift pointing from the barrier. Two-term asymptotic formulas are also derived. The convergence of prices, sensitivities and the first two terms of asymptotics in Carr's randomization algorithm is proved. Finally, it is proved that, in each case, and for any $m\in Z_+$, the error of Carr's randomization approximation can be represented in the form $\sum_{j=1}^m c_j(T,x)N^{-j} +O(N^{-j})$, where $N$ is the number of time steps. This justifies not only Richardson extrapolation but extrapolations of higher order as well.