Hi Meir,
I didn't thought that you were using a PLL. Our spectrum plot were taken without using it. Can you try shielding the power supply or turning off some other equipment in your lab that you are using. In my experience, PLL components don't usually affect the spurs within the spectrum. The capacitor values will affect the phase noise but not much with the spurs. However, interference does and that might be caused by the equipment or even power supplies. Try doing these things first then come back to us.
Best Regards
Louijie