We analyze in this paper a class of explicit exponential methods for the time integration of stiff differential problems. Precisely, we considered Adams exponential integrators with general varying stabilizers. General stabilization brings flexibility and computational facilities for ODE systems and for semilinear evolution PDEs coupled with ODE systems. Stability and convergence are proven, by introducing a new framework that extends multistep linear methods. Dahlquist stability is numerically investigated. A(α)-stability is observed under a condition on the stabilizer, which is a singular property for explicit schemes. The methods are numerically studied for two stiff models in electrophysiology. Its performances are compared with several classical methods. We conclude that for stiff ODE systems, it provides a cheaper way to compute accurate solutions at large time steps than implicit solvers.