However, most likely the company giving you the job will not be able to hire you through your LLC for a period more than a year. This is usually the time period the IRS would rule that you are an employee of the hiring company. This isn't a hard rule however, but a general interpretation. You would of course need cooperation from the company's owners as they may not like having you as an LLC. Even so, there are many people acting and working like employees that are being paid through their own companies.
It will probably be easiest for you to join the new startup as an employee and keep your LLC group coverage. Be careful that you are not eligible on two group policies as in some cases they could cancel each other out. Also try to maintain some income to the LLC to show it is actually trying to make some money and is not a shelter for your hobbies. At least enough to cover expenses.
Also, there is always hope you can again get insurance. Try posting your specifics on this agent forum and see what your options are:
www.insurance-forums.net