BE58 ENTERS MIL RESTR AREA WITHOUT CLRNC WHILE DEVIATING WX IN ZJX AIRSPACE.

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CLRED DIRECT ALMA ENRTE AVL IFR 8000 FT. TSTMS ON AIRBORNE RADAR AND STORM SCOPE. DEV REQUESTED DIRECT CRAIG; DIRECT SAVANNA. TOLD 'UNABLE; CONTACT NEXT SECTOR.' GIVEN 3 NEW FREQS. NONE AWARE OF MY ACFT. BEGAN 360 DEG TURNS TO AVOID TSTMS AS REQUESTED HELP. GIVEN BACK TO INITIAL CTR CTLR WHO ACCUSED ME OF VIOLATING MIL 'HOT' AREA. LOTS OF COMPLAINING ABOUT 360 DEG TURN; THEN GIVEN INITIAL REQUEST DIRECT CRAIG. I BELIEVE THE SIT OCCURRED DUE TO BUSY CTLRS 1) FAILURE TO PROVIDE CLRNC FOR WX DEV; 2) FAILURE TO GIVE PROPER FREQ FOR HDOF; AND 3) URGENT NEED TO DUMP REQUEST TO NEXT SECTOR RATHER THAN DEAL WITH THE PROB INITIALLY. IT IS UNCLR WHY HE WOULDN'T ALLOW DEV FOR WX INITIALLY BUT WAS ABLE TO AFTER I TALKED TO 3 OTHER CTLRS. FACTORS LEADING TO THE OCCURRENCE WERE: 1) LARGE AMOUNT OF TFC. 2) WX MOVING EASTWARD SLOWER THAN FORECAST. 3) NARROW CORRIDOR TO DEVIATE. POSSIBLE MECHANISMS FOR PREVENTION INCLUDE: 1) MORE ACCURATE HDOF. 2) BETTER ATC EQUIP TO REDUCE CTLR WORKLOAD. 3) BETTER CTLR UNDERSTANDING OF NEED FOR WX AVOIDANCE IN A TIMELY FASHION WHEN TSTMS ARE CALLED. THIS HAS HAPPENED BEFORE IN THIS SECTOR.