The private sector grew, the public sector shrank and Douglas County’s unemployment ticked up in July, the Oregon Employment Department announced Tuesday. The county’s jobless rate of 12.3 percent was slightly higher than June’s 12.1 percent, even though private-sector employment grew by 230 jobs. Most of the added jobs were in construction and in the broad category of professional and business services. Those job gains were more than offset by the net loss of 670 public-sector jobs, even though the federal government added 50 positions. Public-sector job losses were driven by a seasonal decline in education positions. Employment in local …