You use the terms "new job", "my boss" and "hourly wage" so you think you are an employee but they are treating you as a sub contractor who is responsible for their own taxes. Are you an employee ? Or are you self employed ?
See what the IRS says ... and get on the same page as this "employer/client" now before you go too far ...
https://www.irs.gov/taxtopics/tc762
