Will you have to shut down your site?
No.
Why?
You can host in the UK, if you are worried about hosting in the US.
Domains registered.. if they are with a registrar in the US, records from registrar can be subpoenaed, but if they are with a UK register.. This won’t happen. (Seen it happen)
You can also put a big fat disclaimer on your page, if you are a citizen of these countries, or these states.. You are not allowed to gamble online, and can only use the website for information purposes. (Just like with adult)
alternatives, sell your domains to me.