I think this may continue for some time. I could say more, but I don't want to get too political.
If the shutdown is still in effect, we won't camp on federal land
If any areas are outright closed, then obviously nobody should ignore the closures. If they are still officially open, on the other hand, I don't have a problem going there and even camping during the shutdown.
The way I look at it, it's public land just like our streets, highways, and public buildings. Clearly, from time to time, public places have to be closed for various reasons. I can understand the closure of certain developed recreational facilities or sensitive natural or archeological areas where the lack of staff during the shutdown could result in sanitation concerns, vandalism or other crime, or ecological damage. Where I would be unhappy would be if there was a blanket closure imposed for no valid reason or as a political response.
Most of the facilities in the Shawnee, including self-service campgrounds, really are not going to be much affected unless the shutdown drags on for months. This is especially true if people realize there aren't going to be normal services like trash, water, restroom cleaning, or mowing. Those who approach it as if it were a step up from primitive dispersed camping won't have any problems. Pack it in, pack it out because there won't be anyone payed to pick up litter. Going the extra step of packing out the trash left behind by others, as you advocate, is good. If the public lands can be kept as tidy and undamaged as possible during the shutdown, that will take away most reasons for closure.