Upvote:3
TLSContact is definitely wrong, it's perfectly possible to cancel (technically: βrevokeβ) a visa, without prejudice, if the holder asks for it (and also for a number of other reasons). Maybe they can't do it themselves but the consulate could. Once your current visa is revoked, you would be able to follow the usual procedure so that would indeed be a solution.
But the Greece visa center is also wrong and the French consulate is right: It is possible to issue a visa starting immediately after the last day of validity of your current visa and to use both visas together for a single trip (see e.g. Schengen visa "refresh" and Traveling on back-to-back Schengen visas). In your case, Greece could therefore issue a visa with validity beginning on September 21 without doing anything to your current visa and that would be another solution.
So both consulates have a solution available but they try to get rid of the problem and want you to get the other consulate to deal with it. Concretely, you need to convince one of them to budge. You could for example approach the Greek consulate/visa centre again, this time with a printed copy of the French consulate's opinion to bolster your case and show them that you can't easily have your visa cancelled. Be calm and polite but firm, in writing or in person, as applicable.