(I’ve name-changed for this because it’ll have some identifying information that I don’t want linked to my main username.)
OP, I’ve written a PhD which deals with this question. You’re not being at all unreasonable. Welshness is a complicated idea, and there are a lot of competing definitions. It’s about ancestry, or it’s about the language, or it’s about cultural traditions, or it’s about a connection to the land, or it’s about nationalism, or it’s about rugby, or it’s about rejecting British imperialism, or it’s about engaging with your local community…
Welsh people have had to fight for centuries to maintain a distinctive cultural identity, and for that identity to be treated as valuable - so there can be this weird feeling that if you’re just quietly going about your life you’re not “doing enough” to deserve to claim Welshness.
If you feel unsettled or distressed by this feeling of not being “Welsh enough”, I’d suggest trying to work out where the feeling comes from. If it comes from other people making you feel inferior, then they’re dicks and they can do one. I’m angry on your behalf if that’s the case. But if it’s coming from your own mind, maybe it’s the expression of something that you feel is missing from your life? If so, you could think about ways to engage with your Welsh identity. If you’re artistic, explore Welsh artistic and craft traditions. If you’re into music, check out the Welsh music scene (or go the traditional route and join a choir!). If you’re political, check out local or national groups working on issues that matter to you.
If you have the time to read, there are some great books that explore this question of what Welshness means. I’d really recommend an essay collection called “Welsh (plural): essays on the future of Wales”, and also “Sugar and Slate” by Charlotte Williams.