The lunar month is not dependent on the distance to earth by itself. Rather it's a consequense of the relative positions of earth sun and moon and how they evolve through time. So an adventurer in your campaign would not be able to identify the, say, epoch just by noticing the odd lunar calendar. At least not without more context. Also I'm pretty sure the long term drift of the moon would be towards earth rather than away from it, even if these kinds of extrapolations are not reliable since they presume that things we've observed in the last milennia or so will hold true for several many more to come.
What you can use reliably to hint at a distant future using astronomy is the expected evolution of the celestial chart, caused mainly by precession and the movement of the sun along its galactic orbit, or the spectrum and apparent size of the sun, since it will turn redder with time. It depends on how far into the future ou want to go. But then again nobody is going to notice without appropriate context.
Good luck in your campaign! I'm a dm too I love worldbuilding.