LIGHT TWIN FLT CREW DEPARTS GAI INTO TRAFFIC PATTERN NOT KNOWING THEY ARE IN DCA ADIZ.

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WE LANDED IN GAI TO PICK UP PAX TO TRANSPORT THEM TO ZZZ. THE RWY AT GAI IS ONLY 4200 FT SO TKOF PERFORMANCE WAS AN ISSUE. I TOLD THE LINEMAN TO PUT 30 GALS IN THE ACFT. HE THEN FILLED THE PLANE UP AND PUT 204 GALS IN IT. NEITHER CREW MEMBER WAS STANDING BY TO SUPERVISE THE FUELING. WHEN WE FOUND OUT THE PLANE WAS OVER-FUELED; THE PAX HAD ARRIVED. I CALLED THE COMPANY AND SPOKE TO THE CHIEF PLT AND WE DECIDED; WITH THE DIRECTOR OF OPS' APPROVAL; THAT THE FASTEST WAY TO GET RID OF THE EXCESS FUEL WAS TO BURN IT OFF. I INSTRUCTED THE OTHER CREW MEMBER TO PREPARE THE PLANE FOR FLT AND BRIEFED HIM ON WHAT WE WERE GOING TO DO. I CHKED THE CHART ON OUR EFB USING FLT PLANNING SOFTWARE AND NOTED THAT THE CLASS B AIRSPACE FLOOR WAS AT 3500 FT. WITH GAI AT APPROX 500 FT ELEVATION I SELECTED 2000 FT MSL AS A GOOD TFC PATTERN. I WANTED TO STAY IN THE TFC PATTERN AS I KNOW WASHINGTON WAS TO THE S AND I WASN'T TOO FAMILIAR WITH THIS AREA FOR VFR FLT. I FAILED TO NOTICE THE ADIZ ARC FOR WASHINGTON DC WAS OUTSIDE OUR LOCATION; MEANING WE WERE IN THE ADIZ AREA. ON TAXI OUT THE FO DID EXPRESS CONCERN THAT WE WERE SQUAWKING 1200 (VFR CODE) AND I DID NOT PURSUE THIS; TELLING HIM IT'S OK AS WE WERE NOT GOING NEAR THE CTLED AIRSPACE. HE THEN LOOKED AT THE ARPT DIAGRAM AND CONCLUDED I WAS RIGHT. WE DEPARTED RWY 14 TO MAKE L TFC AND STAY IN THE PATTERN; OVER-FLYING THE FIELD ON EACH UPWIND. WE WERE MAKING POS RPTS ON THE CTAF. ON THE SECOND CIRCUIT; THE FIELD UNICOM CALLED US AND ASKED US OUR SQUAWK CODE. WE TOLD HIM 1200. UNICOM LATER CALLED AND INFORMED US THAT POTOMAC APCH HAD CALLED AND WANTS US UP ON THEIR FREQ. AT THIS POINT I REALIZED WE WERE IN TROUBLE AND I TOOK OVER THE RADIO DUTIES AND SWITCHED OVER TO POTOMAC AND CALLED THEM. THEY INFORMED ME THAT WE WERE IN THE ADIZ. I ASKED WHAT THEY WANTED US TO DO AND THEY SAID TO LAND AND GAVE ME A PHONE NUMBER TO CALL ON THE GND. WE WERE IN THE DOWNWIND TO BASE AND MADE A LNDG FROM THERE. WE TAXIED INTO THE RAMP AND I CALLED THE NUMBER POTOMAC HAD GIVEN ME. THE FBO INFORMED ME THAT THE SECRET SVC WAS ON THE WAY. THE LCL PD AND SECRET SVC SHOWED UP AND THE SECRET SVC INTERVIEWED MYSELF AND THE FO. THERE WERE MANY FACTORS LEADING UP TO THIS EVENT. MY FAILURE TO NOTICE THE ADIZ ON THE CHART BEING PRIMARY. OUR COMPANY HAS A PROC TO PUT AN ADIZ WARNING ON OUR TRIP SHEETS FOR ARPTS IN THE WASHINGTON DC ADIZ AND FOR SOME REASON THIS WAS MISSING ON MY TRIP SHEET FOR THE DAY'S FLTS. MY FO RIGHTLY EXPRESSED CONCERN ABOUT OUR VFR SQUAWK CODE AND I FAILED TO PURSUE THAT. AS A CREW WE FAILED TO MONITOR THE FUELING OP IN THE FIRST PLACE. WE GOT HURRIED AS BURNING THE FUEL WAS GOING TO PUT US BEHIND SCHEDULE AND OUR PAX WERE ALREADY WAITING. AS I SAID; THE ERROR CHAIN WAS LONG; HOWEVER; AS PIC I FAILED TO BREAK THE ERROR CHAIN AND ALLOWED THE OP TO CONTINUE THINKING I HAD IT ALL UNDER CTL. ON A PERSONAL NOTE; THIS HAS BEEN A REAL EYE OPENER AND I FEEL AS THOUGH I NEED TO RE-EXAMINE HOW I RUN MY COCKPIT TO BETTER UTILIZE ALL INFO AND BE MORE INFORMED ABOUT EACH AND EVERY FLT; ESPECIALLY THOSE IN ABNORMAL CONDITIONS LIKE VFR FLT.