B737 CREW FOLLOW TCASII TO AVOID UNCTLED TFC. COMPLAINS OF COCKPIT PRESENTATION OF TCASII TFC.

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I HAD A TOTAL OF 5 HRS SLEEP. SINCE WE HAD TO FLY ONLY 1 LEG; I FELT EVEN THOUGH I HAD NOT RECEIVED MUCH SLEEP AND WAS TIRED; THAT I COULD COMPLETE MY TRIP. DURING OUR CLB TO OUR ASSIGNED ALT OF 10000 FT; MY FO NOTICED AN ACFT ON TCASII AT OUR 10 O'CLOCK POS; ABOUT 3 OR 4 MI; CONVERGING AND DSNDING THROUGH 11000 FT. AT THIS POINT; I WAS HAND-FLYING THE AIRPLANE AND CLBING OUT OF ABOUT 9200 FT. ABOUT 10 SECONDS LATER; DEP CTL ADVISED US OF THE TFC; AND TOLD US IT WAS A VFR TARGET THAT HE WAS NOT TALKING TO. I BEGAN TO SLOW OUR RATE OF CLB TO ABOUT 300 FPM. SHORTLY THEREAFTER; OUR TCASII ANNOUNCED 'TFC; TFC.' MY FO SAID HE HAD THE SINGLE ENG CESSNA IN SIGHT. WE WERE NOW OUT OF 9600 FT AND CONVERGING VERTLY AND HORIZLY; ALTHOUGH IT APPEARED THAT WE WOULD PASS IN FRONT OF AND BELOW THE CESSNA. PERHAPS BECAUSE OF MY TIREDNESS IT NEVER OCCURRED TO ME TO ASK ATC IF WE COULD TEMPORARILY LEVEL OFF AT 9500 FT. IN THE BACK OF MY MIND; NOT WANTING TO DEVIATE FROM AN ATC CLRNC TO MAINTAIN 10000 FT BY LEVELING OFF AT 9600 FT (FAA VIOLATION); I CONTINUED OUR CLB AT 100 FPM OUT OF 9600 FT; UNTIL UNFORTUNATELY; OUR TCASII ALERT TURNED INTO A TCASII RA. OUR TCASII ANNOUNCED; 'MONITOR VERT SPD; MONITOR VERT SPD.' I STAYED OUT OF THE RED ARC AND BASICALLY LEVELED OFF AT ABOUT 9700 FT UNTIL THE TCASII ANNOUNCED 'CLR OF CONFLICT.' ONCE LEVEL AT CRUISE; I ASKED MY FO HOW CLOSE WE GOT TO THE CESSNA. I DO NOT RECALL HOW CLOSE WE GOT AS I WAS CONCENTRATING ON THE COCKPIT INSTS DURING OUR RA. HE STATED THAT WHEN THE CESSNA WAS AT OUR 6 O'CLOCK POS AND ABOUT 1 MI; HE RECALLED THE MODE C READOUT FROM THE VFR CESSNA AT 10400 FT; AND OUR ALT AT ABOUT 9700 FT. PERHAPS A CONTRIBUTING FACTOR MAY HAVE BEEN THAT IN OUR B737-300 WITH EHSI-EADI; THE LOWER L QUADRANT OF MY EHSI IS BLOCKED BY MY CTL COLUMN WHEN THE SEAT IS IN A NORMAL POS FOR FLYING. I MUST EITHER LEAN FORWARD AND TO THE SIDE; OR LOOK AT THE FO'S EHSI. IT IS IMPORTANT TO NOTE THAT WHEN I SAY THAT THE LOWER L CORNER OF THE DISPLAY IS OBSTRUCTED; IT IS ONLY FROM THE CAPT'S SEAT. CONVERSELY; WHEN SEATED IN THE R SEAT; THE LOWER R SIDE OF THE EHSI IS BLOCKED FROM THE FO'S VIEW. AS I HAD MENTIONED EARLIER; THE TFC WAS APCHING FROM OUR 10 O'CLOCK POS; WHICH ON MY EHSI AS OBSCURED (DEPENDING ON THE RANGE SETTING SELECTED). MY FO WAS THE ONE WHO CALLED OUT THE APCHING TFC. THE 10 O'CLOCK POS ON HIS EHSI (AGAIN DEPENDING ON THE RANGE SETTING SELECTED) WAS NOT OBSCURED. SINCE ATC HAD JUST CLRED US DIRECT TO A VOR ABOUT 80 MI AWAY; WE HAD BOTH SELECTED A HIGHER RANGE. PLEASE DO NOT MISUNDERSTAND ME. I REALLY APPRECIATE THE FACT THAT OUR TFC IS DISPLAYED ON OUR ESHI INSTEAD OF OUR IVSI. IT GREATLY REDUCES SCAN TIME WHEN ACQUIRING TFC; BUT IN THIS CASE THERE IS A DRAWBACK TO THIS PARTICULAR SIT. I CAME OFF FLYING THE B777 AS A FO; AND GREATLY APPRECIATED THE EHSI AND EADI SITUATED NEXT TO EACH OTHER INSTEAD OF BEING STACKED ON TOP OF EACH OTHER. MY FINAL WORD IS THAT HAD I SEEN THE TCASII TFC DISPLAY IN THE LOWER L QUADRANT EARLIER; I MAY HAVE BEEN MORE ALERT AS TO HIS PRESENCE; AND HAVE ASKED ATC FOR A LEVELOFF AT 9500 FT. INSTEAD; THE TFC APPEARED TO 'POP UP' AND I HAD VERY LITTLE TIME TO FLY THE AIRPLANE AND THINK ABOUT HOW I WAS GOING TO SOLVE THIS TFC CONFLICT. I AM NOT ABSOLVING MYSELF OF MY RESPONSIBILITY; I COULD HAVE DONE A BETTER JOB AND HAVE LEARNED FROM THIS EXPERIENCE.