I would go with:
I will be overseas next week.
Overseas, an adverb, means to go beyond the seas. Even with an 's' at the end, I think it still functions as a singular (confirmation anyone?). The number of seas doesn't make a difference, and one sea doesn't mean you have to use oversea. See this link for definitions:
However, Merriam Webster also lists oversea, mainly British usage, as having the same meaning as overseas. Acoording to them oversea was used first, in the 12th century, compared to overseas, in 1533. Nowadays, without offence to the Brits, you hardly hear oversea used.
The nearest synonym to overseas is abroad.
I will be abroad next week.
An occasion when you might use over sea (note separation):
They travelled over sea and land.
which, to my ear, sounds better than:
They travelled over seas and lands.
The other issue is whether overseas would apply if you were travelling to another country without crossing any sea or seas.
By the way, don't confuse overseas with oversee, which means to supervise or to watch over.