The best and really only solution is to call the Ranger Station and talk to the local employees. I am a retired USFS Recreation & Wilderness program manager and I can tell you that different National Forests have different approaches to how they manage their recreation sites. Heck, even on a single national forests, two different Ranger Districts may manage things differently.
Here is some general information: Most national forests have their campgrounds open after the "normal use season" but there will usually not be a campground host after Labor Day. Depending upon the area of the country, the water supply will be turned off, garbage bins no longer available, etc. In areas of the country that get snow, many USFS campgrounds are available for camping until the snow builds up (since the campgrounds and sometimes the access roads to the campgrounds are usually not plowed).
Some National Forests manage their campgrounds with their own Forest Service employees. Others contract out the management to "concessionaires" who run the campground, do the maintenance, collect fees, and pay a small percentage to the federal government. When a concessionaire runs the campground, the season can go longer or shorter, depending on when the concessionaire determines that it is no longer profitable for them to keep it open. Sometimes concession-run campgrounds are NOT available for camping after the normal use season, because the concessionaire doesn't want to be liable for use and damage (that they will have to fix) when the campground isn't formally open. But this is not always the case, as some concession-run campgrounds stay open for use, without water, garbage, etc. after the normal season.
Sorry for the long answer, but again, your only sure way to know is to call the Ranger District where you want to camp. And happy camping! This time of year and into October is my favorite time to camp. Beautiful colors, and we have the campgrounds and lakes and woods practically to ourselves. No screaming kids, barking dogs, loud music, or noisy generators right next to you!