Hi there,
I take Dicloxacillin all the time, I'm not sure about the other one you mentioned.
To cut a long story short, Dicloxacillin is NOT for pseudomonas, it's effective against bacteria such as Staph Aureus (assuming the Staph is sensitive to the penicillins ie not MRSA).
Maybe your doc prescribed the Dicloxacillin for Staph, maybe you have cultured both Staph and pseudomonas (very common for CF) and he didn't mention it to you.
Pseudo (the pseudo that usually resides in CF lungs that is) is difficult to treat by oral antibiotics, the most effective oral med is Ciprofloxacin (Cipro), the others being IV or in inhaled forms such as Tobi or even Piperacillin. I'd have a word to your doctor to check and discuss your treatment.
Good luck