My web server is (include version): Ubuntu 16.04
I can login to a root shell on my machine (yes or no, or I don't know): Yes
My website uses Nginx.
My APIs for mobile apps are not working on Android 6 and giving error: javax.net.ssl.SSLHandshakeException: java.security.cert.CertPathValidatorException: Trust anchor for certification path not found.
Above android 6, APIs are working fine.
sorry but don't think many options left.
This is to be expected and was announced almost a year ago.
One option to consider it trying other ACME CAs such as BuyPass or ZeroSSL: ACME Certificate Authorities - there are several and some may be using a root certificate that's still compatible with Android 6, you would need to test which one meets your requirements and update your certificates using that CA.
In addition to ancient clients, your server OS has been unsupported for almost three years. You should seriously look at updates on both ends.