Can I apply for citizenship on my own?

An adult age 21 or older who has been a Permanent Resident (has had a "Green Card") for more than 5 years generally may be eligible to apply to become a naturalized citizen, regardless of whether she may have become separated from her U.S. citizen husband through which she applied to become a Permanent Resident. It then would not be necessary to supply records of a joint bank account or jointly filed taxes (but it may be necessary to show documentation that you and your husband were living in a bona fide marriage before you became separated). It would be wise for you to consult with an immigration attorney to examine all of the relevant details and advise you about eligibility and to represent you in the application process. That attorney also could address whether your children, who became Permanent Residents before they reached age 18, might qualify to automatically become U.S. citizens without having to go through the naturalization application process (although they would need to file an application for a Certificate of Citizenship if they were eligible and they wanted documentary evidence of their citizenship). Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.

It seems you need to file for divorce; yes you can file on your own and do not need to present evidence of viable marriage.