Nonthermal radiation from supernova remnants (SNRs) provides observational evidence and constraints on the diffusive shock acceleration (DSA) hypothesis for the origins of Galactic cosmic rays (CRs). Recently it has been recognized that a variety of plasma wave-particle interactions operate at astrophysical shocks and the detailed outcomes of DSA are governed by their complex and nonlinear interrelationships. Here we calculate the energy spectra of CR protons and electrons accelerated at Type Ia SNRs, using time-dependent, DSA simulations with phenomenological models for magnetic field amplification due to CR streaming instabilities, Alf$acute{e}$enic drift, and free escape boundary. We show that, if scattering centers drift with the Alf$acute{e}$en speed in the amplified magnetic fields, the CR energy spectrum is steepened and the acceleration efficiency is significantly reduced at strong CR modified SNR shocks. Even with fast Afv$acute{e}$nic drift, DSA can still be efficient enough to develop a substantial shock precursor due to CR pressure feedback and convert about 20-30% of the SN explosion energy into CRs. Since the high energy end of the CR proton spectrum is composed of the particles that are injected in the early stages, in order to predict nonthermal emissions, especially in X-ray and ${gamma}-ray$ bands, it is important to follow the time dependent evolution of the shock dynamics, CR injection process, magnetic field amplification, and particle escape. Thus it is crucial to understand the details of these plasma interactions associated with collisionless shocks in successful modeling of nonlinear DSA.