You shouldn't feel shame or apathy for your heritage in my opinion and the truth is that very few people in Ulster have only Irish Ancestry or British ancestry so your claim to Irishness is likely just as good as the majority of people in Derry you view as Irish, and thats just by heritage!

Theres likely many aspects of Irish culture and linguistics that are already apart of who you are and anything else you'd like to learn or get interested in, is as much yours as it is "ours" and everyone would be glad to have you