you can list your current job as a reference when you apply for a new job. not having matching SSNs wont matter because when a new or prospective employer calls a reference, they usually ask how long as the person worked there, what kind of employee they are, etc. nothing gets asked about SSNs. if you work for a company that has an HR department, then ask the HR department what needs to be done in order for you to change your SSN. this comes with some risk as well. if you lied about being able to legally work in the US and used a false SSN, then the company does have the right to terminate you due to the fact that you lied on your application. i'm not saying this to scare you but just to prepare you for what could happen. it's always good to have a plan B in case this happens.
if your CNA license is through the state, like it is here in CA, i would call the state department that handles CNA licenses and ask how you go about changing your SSN on your license. they may ask why you need to do so and if you applied with a false SSN, then that could present a problem.