Is that what I can expect from now on? Re-wetting the sponge every few days?
Maybe not.
I suspect you could get away with only wetting it at certain times of the year.
I'm not familiar with Austin's range, but the ideal range for acoustic guitars is 45%-55% relative humidity.
If Austin's in that range for the best part of the year, then I'd only worry about humidifying seasonally, if it falls below 40-45%.
Here in the US, too
low is normally a bigger worry than too high, as solid woods can and do crack when they get too dry.
Members in the Northeast for example typically get away without humidifying in the spring/summer/fall months then start humidifying in winter when the air gets dry (the colder the air, the less moisture it can hold) and is even further dried out by typical home heating systems.
Humidity changes can also affect the geometry of the guitar and even set-up as the topwood expands and contracts.
I can recall one member who mentioned he even swaps out a "winter" and "summer" saddle to compensate for the seasonal changes.
Anyway we've seen real cheap digital home hygrometers before, one of those should be able to keep you informed about what's going in inside your house in plenty of time to take corrective action if needed.
It takes a while for damage to be done: several days at least, if not weeks.