ZDC Center Controller reported they did not realize they climbed an aircraft through the altitude of head on traffic until the Conflict Alert activated resulting in an NMAC. The reporter stated they were working a high volume of traffic in an unusual combination of three sectors due to staffing problems in the area.

Narrative

I was the Radar Controller on Sector XX; XY; XZ. Not a normal sector configuration. Aircraft X southbound on v29 at 8;000 feet. It was given a clearance to climb to 10;000 feet. 17 minutes away from Aircraft Y which it was northbound on v29 level at 9;000 feet. Aircraft X is at 9;100 feet. And 3 miles from the head on Aircraft Y it was at this moment that the conflict alert went off. Then I proceed to issue a traffic alert to Aircraft X with no response. A second traffic alert was made to the same aircraft. A regular traffic call was made to Aircraft Y. Aircraft are on top of each other separated by 500 feet. This event happened because of the high traffic volume; complexity of the sector configuration and negligence. Help was called but there was no Supervisor or CIC in the area minutes before or after the incident. Fellow controller sat down on the D side position to help with the workload. Even with the help the sector was way over saturated with too many data blocks blocking the view of Aircraft X and Aircraft Y.Supervisors or CICs have to be aware of whats going on in the area overall. Also management have to staff the area fully and if that cannot be accomplished; find a way to mitigate the traffic volume through the sectors. Because you cannot keep employees working for long periods of time or making weird sectors configuration by compromising safety. If there is no staffing you have to slow down the traffic it is what it is. Safety is the number one description of this job and Traffic Management Unit and command center are not doing much to keep that.