The modest spin-orbit coupling arising from the n-pi* transitions of phenazine complexes offers the unique means to activate the dual emission of singlets and triplets, i.e., fluorescence and phosphorescence, in organic light-emitting diodes (OLEDs) at room temperature. By spectrally resolving OLED electroluminescence, we investigate the optically detected magnetic resonance (ODMR) of such a metal-free dual-emitting host-guest OLED in separate fluorescence and phosphorescence channels. The phosphorescence component of ODMR exhibits two distinctive resonance lineshapes, which differ between the in-phase and quadrature channels. To rationalize these features, we devise a comprehensive quantum stochastic model relating the phase-sensitive components of the lock-in-detected ODMR signal to the forward and reverse transfer of triplets between the host molecules of the OLED, where they are dark, and the guest molecules, from which phosphorescence occurs. Within the model proposed, the observed resonance lines result from the interplay of spin-dependent electron-hole polaron-pair (PP) recombination and triplet-exciton-polaron (TEP) reaction processes. The analytical description of the model is based on the stochastic Liouville equation treatment of the PP and TEP spin dynamics. Quantitative analysis is performed by means of numerical simulations exploiting solutions of the corresponding Liouville equations based on Floquet theory. The resulting accurate match to experiments allows us to draw plausible conclusions with regards to the numerical values of parameters that characterize the system, including the phosphorescence lifetime and the polaron hyperfine interaction strengths. Based on the theoretical framework used to describe the phosphorescence ODMR lineshape, we compute the resonance line recorded in the conjugated detection channel, i.e., in the fluorescence. Surprisingly, this analysis reveals a substantial contribution of the TEP mechanism to the singlet fluorescence channel, implying that fluorescent species are generated from triplets in the course of the TEP reaction. We argue that the TEP process occurring in our device is a spin-allowed polaron-assisted upconversion of triplets to singlets, in which free polarons scatter off trapped triplets, losing their energy to transfer triplets to singlets of higher energy.