I've got a bad toothache/headache/stomach ache/ear ache are correct. Yes, you should use an article every time toothache is combined with an adjective (bad). and it's the same situation about the other 'aches' (like headache)?

There are a few differences between BrE an NAmE. A North American might have a headache, a toothache, or a stomachache, but we might not go to a hospital. British English goes to hospital and ends their sentences with a full stop instead of a period. I'm happy I'm not suffering from a headache.

In American English, we use the indefinite article "a" even without adjectives: "I have a toothache," "I have a headache," "I have a stomach ache ."