A ZJX Controller reported they vectored and descended an aircraft in an attempt to avoid a conflict but the heading and descent placed the aircraft into conflict with a third aircraft. The reporter was working a sector containing more aircraft than guidelines deem operationally safe.

Narrative

Worked sector with d-side staffed prior to incoming volume. Miles in trail for PBI was 30; miles in trail for MCO was 10. Multiple MCO arrivals on vectors descending through and around overflights and PBI aircraft that also needed vectors to achieve spacing. Due to complexity and volume; PBI in-trail mileage was not fully accomplished; and in some cases abandoned altogether. Attempted to use a vector of 195 degrees to move Aircraft X between a southbound Aircraft Z at 31000 ft. And the northbound Aircraft Y at 34000 ft. But under-judged the rate at which the Aircraft X turned to that heading.A descent was issued to 29000 ft. To Aircraft X with the heading. Upon realizing the Aircraft X was turning faster and descending slower than anticipated; adjustments to the heading were issued to 175 degrees; then 155 degrees; and a rate of descent restriction was issued to Aircraft X. At closest point; aircraft 1 and 2 were estimated around 4.5nm and 700 ft.; with separation increasing due to a westerly vector issued to Aircraft Y and a more easterly vector to Aircraft X. Unfortunately; the Aircraft X could only be vectored so far before also conflicting with the Aircraft Z flight that was southbound (which was the original traffic I was trying to miss while meeting the FL290 restriction to the next sector for MCO arrivals).D-side did not notice the conflict until after I did due to their workload trying to help get aircraft incoming to the sector at safe altitudes and on headings to facilitate the miles in trail restrictions we had to meet. I should have judged the turn for the Aircraft X better and not relied on anticipated performance for the turn with the current wind conditions. With that said; what caused this situation is a systemic and all-too-often occurring issue at ZJX. Long 'times-on-position' and high workload occur on a daily basis. Complexity increases with weather; and thus; fatigue can set in quickly in the middle of a shift or even a busy session.Traffic Management Unit (TMU) needed to do a better job of passing back miles in trail; which they failed. MCO arrivals should have been moved to other routings to prevent vectors and descents through a red sector; which ultimately was what resulted in this loss of standard separation event. The attitude of 'just working through it' is a detriment to aviation safety; and tossing a d-side on a red sector is too often the magic fix for management to declare the red sector's risks mitigated or handled. This is the second day in a row that I have witnessed a loss of separation at this sector due to overloading and lack of a coherent plan by management or TMU to handle traffic with aviation safety as the priority; not the movement of aircraft; and certainly not the satisfaction of the 'users.' the safety culture of FAA management is poor and results in overwhelmed controllers and overloaded sectors; and ultimately; these events occurring more and more often.