PA28 PLT; ON SIGHT-SEEING TOUR; ALMOST HITS RADIO TWR. PLT DISTR WITH PERSONAL ACTIVITIES IN COCKPIT AND SOME EQUIP PROBS.

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IN PREPARATION FOR A XCOUNTRY FLT; I REVIEWED THE SECTIONAL CHART FOR SAN ANTONIO; BUT FAILED TO NOTICE THE 2049 FT TWR TO THE W OF SAT. I PLANNED A WBOUND DEP; FOLLOWED BY A TURN TO THE N. I DEPARTED RWY 12R. THE TWR INSTRUCTED ME TO FLY RWY HEADING. I WAS THEN HANDED OFF TO DEP CTL AND CLRED FOR 'OWN NAV;' BUT REMAINED IN CONTACT WITH DEP. I LEVELED OFF AT 2000 FT MSL; ON A HDG OF 270 DEGS. WORKLOAD REMAINED HIGH DUE IN PART TO AN APPARENT PRECESSION PROB WITH THE DIRECTIONAL GYROSCOPE. I HAD TO REPEATEDLY SET IT; ONLY TO LOOK BACK TO FIND IT IN ERROR BY 20-30 DEGS. I DID NOT HAVE AS MUCH OF MY ATTN FOCUSED OUT IN FRONT OF THE AIRPLANE AS I SHOULD HAVE. AFTER LOOKING AWAY FOR A FEW SECONDS TO CHK POS AGAINST THE SECTIONAL CHART; I LOOKED UP TO THE STARTLING SIGHT OF THE 2049 FT RADIO TWR AT ABOUT MY 1 O'CLOCK POS; APPROX 1000-2000 FT AWAY. I TURNED ABRUPTLY L. WE PASSED WITHIN ABOUT 1000 FT HORIZ CLRNC. WE WERE NOT ACTUALLY ON DIRECT COLLISION COURSE; BUT WE WERE AWFULLY CLOSE TO IT. IT WAS NOT PARTICULARLY GOOD JUDGEMENT TO FLY 2000 FT MSL ABOVE A HEAVILY POPULATED AREA WHEN I HAD THE OPTION TO CLB HIGHER. ACTIONS OR INACTIONS: LACK OF SUFFICIENT PREFLT REVIEW OF SECTIONAL CHART. THOUGH IT IS NOT THE RESPONSIBILITY OF DEP CTL; IT COULD HAVE ENSURED OBSTRUCTION AVOIDANCE IF THEY HAD PROVIDED AN ADVISORY OF THE 2049 FT TWR IN OUR PATH.