It sounds like you might be an unfortunate victim of the confusion over visa requirements.
The US embassy say you you must have a visa if you have ever been arrested for anything (charged or not).
The Visa Waiver Scheme ignores minor crimes, as long as they do not constitute moral turpitude (see other posts on this subject).
The sad irony is that on the face of it, you could have legitimately entered for a holiday on the Visa Waiver Scheme. But in applying for a visa (which would confer significantly greater rights to visit/stay in the US), your application failed on the 'close ties to the UK' requirement.
A tourism B1/B2 visa is valid for up to 10 years, and allows individual stays of up to 6 months at a time. Such a six month period can be extended to one year. You will find plenty of sites on the web detailing precisely how to get your Green card and residency, starting off with a B1/B2 and then extending it after 3 months in-country etc.
If I were you I'd write to the US embassy, and explain that you simply want to visit for a holiday. Ask them what you can present (given the circumstances) to establish in their eyes, that you intend to return. And ask them if it would be possible for them to issue you with a visa with restrictions on it such that you could not do whatever they apparently fear you might. Perhaps a single trip one month B1/B2 visa? I expect such a visa might still confer rights to extend though, so it might not be possible.
I've said it before and I'll say it again. If the Visa Waiver scheme was not an option for anyone who had ever been arrested, I reckon the Florida Tourism industry would collapse overnight.
p.s. A new passport won't help. The US embassy took your fingerprints.