SF340 CAPT BECAME DISTR WITH WX WATCH AND FORGOT TO CLB AS ASSIGNED BY DEP CTL; RESULTING IN TCASII RA WITH ANOTHER ACFT.

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WE DEPARTED THE JACKY 4 DEP FROM DFW; WHICH IS A W DEP ROUTING. AFTER TKOF; THE WX OVER THE NW DEP GATE WAS SEVERELY COVERED WITH TSTMS; ALL THE WAY TO THE TEXAS PANHANDLE. QUICK THINKING WAS IN ORDER BECAUSE MY PREFLT REVIEW OF WX HAD DETERIORATED RAPIDLY IN A SHORT AMOUNT OF TIME. WE REQUESTED A NE ROUTING. IT TOOK ATC A BIT OF MANEUVERING TO GET US OVER THERE. WE DIRECTLY OVERFLEW DFW TO 'GET ON THE OTHER SIDE OF DFW ARRS.' I HANDED THE FO THE AIRPLANE AND ATC WHILE I DECIDED TO CONSULT WITH DISPATCH REGARDING THE WX SIT. AGAIN; THE WX WAS WORSE THAN I OR DISPATCH HAD EXPECTED. AFTER COMMUNICATING WITH DISPATCH THAT WX OVER THE NE OF DFW WAS PASSABLE; I ASSUMED PNF DUTIES. WE WERE GIVEN CLB FROM 4000 FT TO 6000 FT. BECAUSE OF THE HIGH WORKLOAD SIT DEALING WITH WX; TALKING WITH FLT ATTENDANT TO REMAIN SEATED; POSSIBLE ROUTING CHANGES; MANY DIFFERENT VECTORS BY ATC; ETC; WE FORGOT TO CLB. A FEW MINS LATER; I NOTICED TFC AT 12 O'CLOCK POS AT OUR ALT VIA TCASII. I WATCHED TFC AND TOLD FO TO EXPECT AN RA. WE GOT THE RA SHORTLY AFTER THAT. SINCE WE DIDN'T HAVE TFC IN SIGHT; WE EXECUTED A CLB. WHILE WAITING TO ADVISE ATC; ATC QUICKLY ASKED US WHEN WE WERE GOING TO CLB TO 6000 FT. I ADVISED ATC WE WERE NOW CLBING TO 6000 FT. WE VISUALLY ACQUIRED TFC AND NO FURTHER CONFLICT AROSE. THE SIT AROSE FROM MANY FACTORS: WX THAT HAD WORSENED IN A SHORT AMOUNT OF TIME; A WESTERLY DEP WHEN WE REALLY NEEDED AN EASTERLY DEP; COMMUNICATING WITH FLT ATTENDANT COMMUNICATING TO PAX TO REMAIN SEATED; AND CONFERRING WITH DISPATCH ABOUT WX SIT; COMMUNICATING WITH FO ABOUT SAFEST RTE AROUND WX; AND MAKING SURE DEVS WOULD DEPLETE OUR FUEL; ETC.