"Adre" is what you'll hear native speakers say and increasingly you'll see it written that way, except in more formal writing.

I'm not a linguist, so can't give you a technical answer, but it's just something that some local dialects like to add to the positive forms of the past tense of the verb. It causes a soft mutation. So you can say: "Mi ddysges i Ffrangeg" and "Dysges i Ffrangeg". Both mean "I learnt French." To complicate things further, some people use the "mi" construction but then omit the "mi"! In that case, you still mutate, so "Ddysges i Ffrangeg", "Wnes is swper" etc.

It's called a preverbal article (I had to look that up). I think of it as a "marker word", it marks the verb as positive.