You do not say how you know your SSN was "seen" on the dark Web. And you have not mentioned if any fraudulent activity has resulted from the SSN being seen.
If you suspect tax-related fraud or identity theft:
https://ttlc.intuit.com/questions/1900655-what-to-do-if-you-suspect-tax-related-identity-theft
Most important) Follow the IRS procedures for reporting a fraudulent return, documented here: www.irs.gov/uac/Taxpayer-Guide-to-Identity-Theft
IRS INFORMATION REFERRAL TO REPORT VIOLATION OF TAX LAW
https://www.irs.gov/pub/irs-pdf/f3949a.pdf
If your Social Security number is compromised and you know or suspect you are a victim of tax-related identity theft, take these additional steps:
6-digit IP NUMBER FROM IRS
https://content.govdelivery.com/accounts/USIRS/bulletins/2af48f0?reqfrom=share
to get you new PIN from the IRS
- https://www.irs.gov/identity-theft-fraud-scams/retrieve-your-ip-pin
If your SSN has been exposed, that does not necessarily mean it has been actually used for identity theft. However, there are some things you can do to protect yourself and reduce the risk of becoming a victim. See this page for some ideas.
https://consumer.ftc.gov/features/identity-theft
I would start by contacting each of the three credit bureaus and locking or freezing your credit reports. This should prevent anyone from getting new credit in your name. (If you want to apply for credit, you will have to temporarily lift the freezes.). Each credit bureau offers credit monitoring for a fee (will alert you to issues), but the basic freeze is free.
https://www.nerdwallet.com/article/finance/credit-lock-and-credit-freeze
Some credit cards offer credit monitoring and will send you an alert if a new credit line is opened in your name or if your balance changes significantly.
You can also go to your credit and banking apps and set text alerts to trigger for purchases over a certain dollar amount, that gives you rapid warning if someone tries to make a purchase on your info. (I once got an alert that someone was using my credit card to try and buy something in England. I called the card, blocked the transaction, and changed my card number immediately.)
The last major thing is to make sure that all your important web sites have different passwords. That way, if someone gets your password to a low security site like a blog or comment forum, that does not give them access to your financials.