Dispatcher reported releasing an aircraft to an airport that did not have a 5G AMOC for the aircraft type and did not meet weather requirements.

Narrative

I dispatched a flight to an airport that does not have a 5G AMOC for the Aircraft X aircraft type. The flight's dispatch release was initially planned legal at first but the TAF updated before the aircraft reached point of dispatch and I forgot about the flight and the flight took off with 1/4 SM in the new TAF and therefore was an illegal dispatch. At the last second; I remembered about the flight and I sent frantic ACARS messages and tried to call ATC but it was too late the flight was on the takeoff roll when I tried to call ATC and they took off. The Flight landed at intended destination with no further issues after taking off illegally. I was very task saturated today with weather issues and it caused me to get behind. In my opinion; I was more behind today than I normally am because of weather and maintenance issues. I had 2 Aircraft X flights that were departing the same location to the destination that has no 5G AMOC. When the TAF updated and had 1/4 SM now in the TAF for ETA; I reached out to one of of the flights and made sure they did not pushback from the gate. Because I was task saturated and behind; I got tunnel focused and forgot about the other Aircraft X flight at that same airport going to the same destination. By the time I realized I did not tell that other flight to stop too; it was exactly 10 minutes after ACARS OUT time was recorded. I frantically sent ACARS messages to get them not to takeoff and to call me and as well I tried to call ATC multiple times but for some reason Tower would not answer the phone. Unfortunately the flight took off with two alternates on the release but the flight did not qualify for Exemption because there was no conditional language in the TAF at ETA and as well the main body at ETA read '1/4 SM'. For this reason; it was an illegal dispatch. For the record; at time of planning; the ETA showed 1/2 SM in the TAF at ETA and the flight was planned with 2 alternates in case a new TAF came out with conditional language so that Exemption could be used if need be. I will do a better time of issuing flight crews pertinent safety of flight information such as weather per my sole responsibilities in the role of Aircraft Dispatcher. I will do a better time of increasing situational awareness and keep a better track of my flights to ensure safety of flight is never compromised. I will do a better time of duty/task prioritization and ensure that this issue does not occur again.