ZJX Center Controller reported that they issued a 360 degree turn to a departure which placed it into confliction with another aircraft; and to resolve the confliction; the Controller descend one of the aircraft into an adjacent facility's airspace without coordination.

Narrative

The only hole in a large line of weather was forcing most all aircraft through my sectors. I was asked to work 'one more' MCO lander and if they could resume departures. I agreed with 20 MIT because I knew I would not have to climb head on with arrivals. Unfortunately; I did not know I was about to be handed multiple southbound deviating slow GA aircraft. I had them at 17000; 16000; 14000 ft.; and all needing deviations. I forced my MCO lander down early to get under the props. In the meantime I took handoff on Aircraft Y; an MCO turboprop departure at 12;000. The complexity kept building; and then F11 said I had to spin Aircraft X; the departure who was currently at 12;000 ft. When the aircraft turned; they became in conflict with Aircraft Y; and I did not recognize it. Just prior to conflict alert; my tracker noticed and in a panic; I descended Aircraft X into DAB airspace at 11;000 ft. without a pointout. The aircraft were probably 1 minute from hitting and it was simply my first reaction.I should have been more diligent in my duty to protect the integrity of my sector. I was being asked to do too much and pressured to do more and I caved. We did wind up spinning several aircraft; but I should have taken more proactive actions earlier. I'd recommend systemic changes to help protect sectors impacted by severe weather; but I figure at this point why bother? Pull the tape and look for yourselves. Pull the tape from last Saturday; last Thursday; or any other day when there weren't 100% ideal weather conditions at ZJX. No matter what; the pressure to 'keep them moving' takes precedence over doing the right thing. Controllers are being forced to recognize on position in the heat of the moment that things are going from bad to worse and then dump the burden of no notice holding on their co-workers because nobody with a higher level view is capable or willing to see what's happening in a timely manner. I; for one have learned my lesson and vow to never again allow a sector I'm responsible for to get that way ever again.