Houston ARTCC Controller reported a lack of minutes in trail separation between two aircraft. When he attempted transfer to the adjacent facility the deficiency was noted.

Narrative

Air Carrier X souteastbound A626 with Air Carrier Y in trail. Sector was very busy with very bad rides and lots of traffic. Air Carrier X was only five 5 minutes in front of Air Carrier Y on A626; we need 10 minutes with same mach number or 15 minutes in trail same altitude. Numerous aircraft changing altitudes due to rides. Numerous aircraft on frequency. I was on the D72 position passing and accepting flight plans from Merida Center Caribe Sector. I had passed both aircraft to Merida at the same altitude without noticing the lack of time between aircraft. Merida Caribe was training. The R side was releived and the new R side asked me how are we doing these two aircraft? I immediately called Merida and coordinated Air Carrier Y FL340 WAFDOF [Wrong Altitude for Direction of Flight]. The sector was very busy with requests for rides and traffic. These two aircraft were in a bunch of aircraft all going down A626 at the same time. There were so many updates on altitudes and manual updates for times. I was very; very busy and missed these two aircraft. I believe there was no safety issue; just not enough time between these two aircraft. Recommendation; get better flight processing from the HOST computer. There is not enough time to react to this level of traffic. There are so many updates both from the computer and manual (due to no update from the HOST i.e. timely time updates). We use one strip per aircraft instead of the normal two or three we get with each aircraft; this is because we do not have the room to post all the strips on the board. Slow down the amount of traffic entering this oceanic non-radar sector. There were so many updates on the strips due to lack of HOST time updates; we need better times that are accurate.