You've come to the right place! Sharing our mental health struggles with our parents, especially when we're young, presents two challenges that can combine for a tricky balancing act: 1) having your concerns be taken seriously and not dismissed or attributed to unrelated factors, and 2) telling someone who loves you and protects you that you are dealing with an internal struggle that they can't immediately intervene in.
I appreciate your hesitance to self-diagnose, and I'm not a medical professional, but I do think that many of the feelings and phenomena you're describing sound a lot like depression and certainly resonate with my own experiences with it. Whether you choose to take on that as part of your identity, or as a useful shortcut to get the resources you need, or not at all, is completely up to you.
I'm new to the country so am not very familiar with health regulations here, but I *think* (don't quote me on this) if you are over 18 you should be able to see a GP on your own who can refer you to a counselor if that's something you're interested in pursuing. If you live near a Headspace center, you may also receive care there for free. Many online options also exist.
Focus on yourself for now. Connecting with your father about this shared bond can be a conversation to have later and may be a painful or awkward topic, but can also be a powerful point of common understanding in your relationship going forward. It wasn't until later in life that I had very moving and candid conversations with my dad about his depression and how he felt responsible for mine.
I'm sorry that this burden has fallen to you at a time of life that is already confusing for so many other reasons. Please know that there are many who have come before you (myself included), and that there is light at the end of the tunnel.