Social Security field offices, hearing offices, and the Appeals Council are all operating during the shutdown. Those staffing Social Security offices are not receiving their paychecks during the shutdown. However, they continue to stoically go about their duties.
DDS offices are operating as well. However, DDS does not have funding for consultative exams during the shutdown, so those exams are not being scheduled. This may hold up some claims.
As always, if you can conduct your business online, do so.