Are your shifters 9 speed or 10? While there is a good chance the shifters needs to be flushed out, if shifter clicks but the shifter lever (brake lever) is very difficult to push then it is likely the cables are seized in the housings. The best way to tell is to disconnect the cable from the derailleur and then operate the shifter. If the cable does not move you need to remove the cables at the very least and most likely the housings. If the cable moves but the shifter doesn't click, then it's the shifter. Since the problem exists with both derailleurs, I'm thinking it's cables.