121 experience is definitely preferred, however it may not need to be in the left seat...

If you have an immediate opportunity to get 135 turbine PIC, do it! Get about 1500-2000 (or whatever SWA wants), then find a good regional to get some RJ and 121 experience. You can hang out comfortably as an FO until someone calls. Worst case, you have to wait until you upgrade at the regional.