Origin of drought
In American English, drought with the pronunciation [drout] /draʊt/ is common everywhere in educated speech, and is the usual printed form.
Examples from the Web for drouth
Then I has the steward lug up a lot of cold bottles and I breaks a ten year drouth with a whole glass of fizz water.Shorty McCabe|Sewell Ford
"Thou shalt not steal," is just as good a commandment if it should turn out that the flood was a drouth.The Works of Robert G. Ingersoll, Vol. 8 (of 12)|Robert G. Ingersoll
That State was then suffering intensely from drouth, as she continued to do for some weeks thereafter.What I know of farming:|Horace Greeley
Hunter: in open prairies honey yield abundant when season is favorable; drouth injures it.Texas Honey Plants|C. E. Sanborn
All instinct like the bird in drouth got water out of the end of a jar by throwing in pebbles.Ulysses|James Joyce
Word Origin for drought
Old English drugað, drugoð "drought, dryness, desert," from Proto-Germanic *drugothaz, from Germanic root *dreug- "dry" (cf high/height) with *-itho, Germanic suffix for forming abstract nouns (see -th (2)). Drouth was a Middle English variant continued in Scottish and northern English dialect and in poetry.