ZDC Center Controller reported an aircraft resumed clearance to PAAZZ after deviating for weather but ATC had cleared them to PAACK. The reporter stated the similar named fixes caused the confusion.

Narrative

There was popup thunderstorms everywhere. W122 was exclusive and not taking pointouts. There was weather at ILM that was causing aircraft to deviate towards the warning airspace. I called Jacksonville Metta/Knemo sector and told them I could not take any more aircraft on the east side of weather because they were approaching W122. This caused ZJX to route aircraft into the weather to get them on the west side of the weather which was unsafe. During this ZDC Sector 38 called and told me to hold BWI arrivals due to volume and complexity of their sector. They were well over their volume number and holding LGA and EWR.Aircraft X came over from the north deviating during all of this. I was under the assumption that the aircraft was given deviations and when clear to go direct to PAACK. I noticed the aircraft was heading towards the active warning airspace. I asked if they were still deviating and she said she was given direct PAAZZ. I issued a heading to remain clear of the airspace then issued the previous route. I know this was an honest mistake. Aircraft X understood they were allowed to deviate and then go direct to PAAZZ when clear. I assume the controller that issued the deviation did not catch the readback and Aircraft X heard it wrong. We did not violate the airspace; however both fixes were on the aircraft's route.Volume and complexity made this much more difficult. I suggest renaming one of the fixes to sound less similar to avoid this from happening again. Also some plan should have been in place to not have to route aircraft into the weather to stay clear of the warning airspace. A traffic management plan should have been in place to put aircraft on the back side of weather to keep them clear of the warning airspace and keep the aircraft from being routed into weather which is unsafe.