A submerged isotropic active particle (or droplet) that emits/consumes a chemical and interacts with it to drive flow via diffusio-osmotic slip (or Marangoni effects) can exhibit symmetry-breaking spontaneous motion. We derive a reduced-order model for the slow dynamics of the particle near the threshold for spontaneous motion using a weakly nonlinear expansion, which involves matching a quasi-steady particle-scale solution to an unsteady diffusive remote region. The resulting amplitude equation for the particle velocity includes a term representing the interaction of the particle with its own wake in the remote region, which can be expressed as a time integral over the history of the particle motion, allowing theoretical analysis and efficient numerical simulation of fully three-dimensional problems. We study various cases, including the particle interacting with a force, a wall, other particles and/or other weak perturbations, resulting in linear motion, circular motion, and more exotic dynamics.