It looks like when you were born, your father was a US citizen, since he was born in the US.
However, derivitive citizenship is complex as there are many rules to be checked. You may or may not be a US citizen.
I'd consult an immigration lawyer here in the US to determine that fact.
B1 / B2 visa is much harder to get compared to ESTA. You have active ESTA, don't abuse it. Leave on time, don't come to the US for too long and not too often. You can figure out whether you're a US citizen meanwhile. If you're a citizen, you'll follow a process, likely outside of the US to get passport etc. If you're not a US citizen, you won't violate your ESTA by following the rules. If you want to get a B1 / B2 visa, which I don't recommend in your case, you can do it next time you're back in the UK.
If you're not a US citizen already, there's the following ways to get a green card:
- Employment (you need to find a sponsor)
- Green card lottery (you missed your chance this year, but can try next year)
- Asylum (I don't think you can claim it based on the fact you're from the UK)
- Marriage to a US citizen (has to be legitimate one, not for immigration)
- Sponsorship by your parents (will take a while give you're over 21)