I cannot tell you for sure, but I think that you can likely change the core and possibly fix the problem.

I would pick up a couple of new cores, a set of metal stem caps with gaskets in them, and a tool to remove a core.

Then go to some place where air is available.

Remove the core and insert a new one, add the proper pressure and check for a stem leak. I use some saliva to do so any time I check my tires. You DO check your tires?

If the stem still leaks, add a couple of extra pounds and put the metal cap on.

Inspect the core that you took out, the small rubber gasket is probably torn.

OR/THEN

Go to a REALl tire shop and ask them what you should do.My shop would probably help you for nothing. I have been going to the same, family owned shop since 1963, I think that their service is worth a few bucks when I buy tires.
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'96 855R,'64 PV544 driver, '67 P1800 basket case, '95 855, '95 854, the first three are mine, heh, heh, 485,000 miles put on 9 bricks