I recently rebuilt a few shifters and was able to get all my cables loose using a chain / cable lube. Try using masking or painters tape around the end of each end of the cable to form a cone at the enternce of the cable. Fill that with the lube and work the cable back and forth removing gunk from the cable during the process. Unless the cable itself is compromised this should work. I suggest taking it completely apart before you buy a new cable. Either way, be sure to fill the shifter with grease upon reassemblely. Some of the ones I took apart had none remaining and dirt/grime in its place.
Here are other things to consider:
The shifter itself could be worn. As you can see in this picture the shifter grears/stops are worn.
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And in this picture the pull shift mechanism failed on the left 2 pieces. One the nipple is sheared right off and the other the spring stop has let go and no longer working.
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