A GA PLT RPTS THAT A BUSY ATCT LCL CTLR BECAME TOO BUSY TO WORK HIM FOR A PRACTICE APCH TO RWY 07L AND HE WAS SWITCHED TO THE RWY 07R. AFTER CONTACTING THE CTLR FOR RWY 07R HE WAS ALLOWED TO BREAK OFF THE APCH AND DEPART FOR SJC THROUGH THE CLASS D AIRSPACE.

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I WAS HANDED OFF FROM STOCKTON APCH TO LIVERMORE TWR ON A PRACTICE VFR ILS APCH RWY 25R ABOUT 1 MI PRIOR TO LOM. DUE TO THE VERY BUSY TFC AT LVK I WAS NOT ABLE TO CONTACT TWR UNTIL ABOUT 1 MI PAST LOM INBOUND. WHEN I RECEIVED A REPLY IT WAS VERY RAPID AND WHAT I REMEMBER WAS AT 3 MI TURN L (RWY 07 WAS ACTIVE). DUE TO THE AMOUNT OF TFC AT LVK I CALLED TWR AND REQUESTED TO BREAK OFF THE APCH AND TURN L TO SAN JOSE. THIS WAS ABOUT 3 MI FROM LVK. LVK TWR HANDLING RWY 07L AND THE ILS AND ME. NEVER DID GET MY CALL SIGN CORRECT AND SEEMED CONFUSED AS TO WHO AND WHERE I WAS. HE RESPONDED FOR ME TO TURN R NOW AND CONTACT THE OTHER TWR OPERATOR HANDLING RWY 07R. I CONTACTED HIM AND HE CLRED ME THROUGH THE CLASS D AIRSPACE AND TOWARD SAN JOSE. HE THEN CALLED ME AND WANTED TO KNOW WHO AUTH ME INTO LVK; WHEN WAS I AND WHERE WAS I WHEN SCK APCH CTL HANDED ME OFF TO LVK. HE ALSO WANTED TO KNOW WHAT CLRNC I WAS GIVEN FROM LVK AND WHERE I WAS. I TOLD HIM AND HE THEN SAID THANK YOU AND I ASKED FOR A FREQ CHANGE HE GAVE IT TO ME AND I WENT TO SJC. DUE TO THE HVY TFC LOAD AT LVK AND WITH THE ACTIVE RWYS 07L AND 07R THERE IS NOT MUCH TIME FOR CONVERSATION ESPECIALLY WHEN THERE IS CONFUSION AS TO CALL SIGNS AND WHERE TFC IS LOCATED. I DON'T THINK THERE WAS ANY CONFLICT BUT THERE WAS CONFUSION BY THE TWR AS TO WHO AND WHERE I WAS. EVERYTHING WAS SORTED OUT ONCE I WAS SWITCHED TO THE OTHER TWR OPERATOR.