A lot of owners of my previous device used to have the opposite problem due to the switch inside the headphone jack getting stuck in the pushed-to-one-side position, so the device always thought the headphones were plugged in. It could be cured by repeatedly plugging and unplugging the jack, or by unsticking it with a needle if all else failed. I can't really imagine the switch getting stuck the other way since you wouldn't be able to insert the plug, so maybe it's broken? Alternatively, maybe it's sofware-related - have you tried a battery pull just in case?
I don't think alcohol would be a good idea (at least, not in the jack; I don't suppose it would do the plug any actual harm, although it seems unlikely to help anything).