You should use "anywhere". In this context, "anywhere" would be used for a sentence in which the verb was negative, and "somewhere" would be used for a sentence in which the verb was affirmative. Examples: "I didn't go anywhere" (negative) versus "I went somewhere" (affirmative). Other similar words that follow the same pattern are: "anything" and "something" "anyone" and "someone". Examples: "I don't want anything" (negative) versus "I want something" (affirmative) "I don't know anyone" (negative) versus "I know someone" (affirmative).

I didn't go anywhere except Batumi on my summer vacation. Notes: 1. I added an apostrophe in 'didn't' 2. We go 'on' vacation, not 'in.'

"Anywhere" would be used here.

It should be 'anywhere', as you have written, alternatively you could say 'I went only to Batumi in my summer vacation.