There are a couple of different ways to think about this. The general approach is applicable in a world of stochastic rates and volatility, you just have to parameterize their processes and solve the optimal stopping time problem in a richer space, and my comment remains correct, mutatis mutandis. Of course, then you will tell me that the parameters of the rate and/or vol dynamics will change, and we are back to where we started. The more pragmatic answer is that you estimate the original model boundary in order to determine the option value and whether it is *currently* optimal to exercise it. If it is not, then you kind of forget about it and repeat the process until you reach the earlier of maturity or a point where you decide that early exercise is indeed optimal. As usual, model recalibration introduces a kind of temporal inconsistency of behavior, but you get used to that after a while (for better or worse).
I am looking at the practical application and wonder if I need to keep the exercise boundary out of my solver at all. Market most often evolves not exactly as implied by the model and therefore at any instance I need to reprice an option and as a result recalibrate my boundary. Assume I am holding an American put and as I am at the next day, now I ask myself again, what is the price and is it optimal to exercise today? I assume intrinsic value is always less then the price, as otherwise I would exercise immediately. So, do I get any use from yesterday's numerical boundary I got from the numerical solver?