ADS LCL CTLR FAILS TO CHANGE A C172 STUDENT AND INSTRUCTOR PLT ON IFR CLRNC TO D10 APCH CTL AFTER TKOF RESULTING IN ACFT PENETRATING DAL DEP CORRIDOR AND REROUTING OF TFC.

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I WAS DEPARTING ADS ON AN IFR FLT PLAN TO DWH. AT APPROX XA25 HRS; I CALLED ADS CLRNC DELIVERY TO PICK UP MY IFR CLRNC. I WAS CLRED TO DWH VIA CLBOUT ON RWY HDG (150 DEGS); THEN RADAR VECTORS TO JOIN THE JOE POOL NINE DEP; BILLEE TRANSITION (JPOOL9.BILEE); THENCE THE HOAGI ONE ARR (HOAGI.HOAGI1). CONTACT DEP ON 124.3 WITH SQUAWK. I HAD CLRNC DELIVERY REPEAT BACK MY CLRNC BECAUSE I DIDN'T CATCH THE FULL DEP PROC BECAUSE I WAS UNABLE TO COPY AS FAST AS THE CLRNC CTLR WAS SPEAKING. THE CTLR REPEATED THE CLRNC AND I REPEATED IT BACK. I GOT THE READBACK CORRECT AND PREPARED TO DEPART. RUN-UP AND TAXI WERE UNEVENTFUL. AFTER TKOF; MY STUDENT CLBED TO 2000 FT MSL WHILE MAINTAINING A RWY HDG OF 150 DEGS. APPROX 3-5 MINS AFTER TKOF; I STILL HAD NOT BEEN HANDED OFF TO REGIONAL APCH CTL. AS WE CLRED DOWNTOWN DALLAS; I REALIZED THAT WE HADN'T BEEN HANDED OFF AND WE NEEDED TO. THE ADS TWR FREQ WAS VERY BUSY AND IT TOOK SOME TIME FOR ME TO BREAK IN. I CALLED TWR AND TOLD THEM WE WERE STILL ON THEIR FREQ AND DID THEY WANT US TO REMAIN WITH THEM? TWR SAID THAT WE SHOULD HAVE CONTACTED REGIONAL APCH CTL ALREADY. I ACKNOWLEDGED AND IMMEDIATELY CHANGED FREQ AND CHKED IN WITH REGIONAL APCH. APCH ASKED WHERE WE WERE AND WHAT MODE C XPONDER CODE WE WERE SQUAWKING. I REPLIED THAT WE WERE JUST N OF DOWNTOWN DALLAS AT 2000 FT MSL SQUAWKING XXXX. APCH STATED WE NEEDED TO SQUAWK XXXY AND THEN TURN IMMEDIATELY TO 030 DEG HDG. WE MADE AN IMMEDIATE L TURN TO 030 DEGS AND SQUAWKED XXXY. A SHORT WHILE LATER; APCH GAVE US A R TURN TO 090 DEGS AND INQUIRED IF WE HAD RADIO PROBS. HE ALSO STATED THAT WE HAD FLOWN INTO DAL'S DEP PATH AND THAT DAL HAD TO SCRAMBLE TO REROUTE SOME JET TFC. I REPLIED THAT ADS TWR NEVER GAVE US FREQ CHANGE AND THAT I HAD SQUAWKED XXXX DUE TO CLRNC GIVING ME THE INCORRECT CODE. I ALSO STATED THAT I HAD WRITTEN DOWN THE CODE XXXX AND HAD READ THAT BACK TO ADS CLRNC DELIVERY. ADS CLRNC DELIVERY TOLD ME THAT MY READBACK WAS CORRECT. WHEN I RELATED THAT INFO TO THE REGIONAL APCH CTLR; HE REPLIED 'THAT EXPLAINS IT.' REGIONAL APCH GAVE US FURTHER CLB AND VECTORS; AND THE REST OF THE FLT TO DWH WAS UNEVENTFUL.