According to Ngram, it is just as common in the US as the UK, and more common than a couple of other terms for weather periods that I thought of for comparison.
My guess is that you live somewhere where it is predominantly dry (of which there are none in the UK). The terms only makes sense if you consider the weather to be wet by default, like they do not have icy spells in the Antarctic as it is usually icy.
I cannot find any etymology for this particular use of the word, which does not seem to relate to the others, nor can I find a date for its first use.
However, the Ngram above suggests it started in about 1850 and grew in a linear fashion until 1940. This corresponds almost exactly with the birth of the telegraph and the consequent development of the modern weather forecast, suggesting that the term was essentially invented for weather-forecast publication.
5 Aug 1887. You're right about the lack of etymology, it's just in the "Special attributive combinations" section of the headword "dry". (The full quotation is "Everybody found smoking on the streets..during the dry spell was liable to be arrested."; their first British one is 1920 "A Dry Spell is a period of fifteen or more consecutive days no one of which is a ‘Wet Day’." from something called British Rainfall 1919.)
Well the first thing I notice is that the OED needs to do a bit more work. Ngram has found usages in the US and the UK long before that.
I get really frustrated with the lack of etymologies for particular uses of a word. The number of times I have thought that a particular use sounds more like a borrowing from the Gaelic than a development of a previous use, only to find the OED and others have not actually considered it as a separate item.