Translation:My cousin and I have a watermelon.
It is correct as it is. Your suggestion, while very popular in everyday speech, is grammatically wrong. :) Since the pronoun is the subject in the main clause, it is like saying "I have a watermelon". You wouldn't say "Me have a watermelon". The fact that there is another subject (my cousin) and an 'and' there makes no difference.
well, americans never say I and my friends or I and my brother. I think τουλαχεςτον modern version should be accepted as well as the correct one )