This theroretical work investigates the properties of nonlinearly-operated weakly-coupled resonators (WCRs) for resonant sensing applications. We propose an analysis framework for actively-coupled (MILOs) or passively-coupled (MOLOs) self-oscillating WCRs, subject to nonlinear restoring and damping forces. Under some simplifying assumptions, three sensor architectures are investigated and compared, highlighting several common features such as : (i) the insensitivity of the amplitude ratio output metric to the A-f effect, (ii) the instability of one oscillation state above a threshold amplitude. These results are illustrated and validated using transient simulations. Their range of validity is then discussed with respect to finite perturbations, finite bandwidth, measurement noise and nonlinear dissipation-fluctuation. Part 2 of this paper proposes an experimental validation of our analysis.