Jane -
You've almost got it.  The simulator does compare the solutions at the start and end of the period, but if the difference is too big, then it doesn't (generally) add a deltaT, it changes the initial point (voltages and currents) and re-computes the transient for a period and compares again.  One could just run the transient analysis for another period (fixed-point iterations), but more sophisticated simulators use "shooting Newton" which uses the circuit matrix at each timepoint along the period to figure out which way to perturb the initial conditions.

When doing PSS for an oscillator, the period can change (deltaT), but also the initial condition can as well.  For many (driven) circuits, the period is known, eg, you know what the local oscillator (LO) frequency is because you have a sine source driving your mixer.

For each timepoint in the periodic solution from t to t+T, the circuit is linearized, and then the response at each timepoint (to a small-signal stimulus or noise) is added up (weighted, I think, by the length of the timestep).