I went ahead and reached out to HR and they were very understanding. My name won't be fully changed before I start. She told me to continue using my married name until I receive my new ID and SS card and I won't be able to submit for my SS card until next week after I've officially started. So it will work out.
All the SSA says is that it is preferred that the court documents used for changing your name are not over two years old. If you wait two years or more to file for a name change, in other words, there can be more hoops for you to jump through.
And of course, if you decide to keep your married name after all, you won’t have to worry about any other changes. You can keep your social security, driver’s license, etc., the same.
The place for you to start, therefore, would be with the court to get the decree amended, so you can reverse the recent legal change of your name.
Good luck with your new job.