If I was charged with assault for domestic violence 3 years ago and was told I would never be able to work as a phlebotomist again, is this correct?

Unfortunately, many people enter pleas to misdemeanor assault charges thinking, "it's just a misdemeanor, it can't hurt that bad..."  Unfortunately, more and more employers are declining to hire people who have any assaultive issues on their criminal history.  With that being said...you are not without options.
The first is to seek employment through a different agency or employer.  Not all with disqualify and with enough references, one may give you an opportunity.  To increase your chances, you may want to apply for certification through one or more of the national phlebotomist organizations.  
 If you continue to be denied employment, you may want to consider visiting with a criminal appellate attorney to review your case.  Emphasis here is "appellate" attorney, not just a regular attorney.  Roughly four years ago, immigrants were able to successfully challenge a large number of convictions on the basis that their original attorneys did not properly explain all of the consequences of their pleas to them regaring their immigration status... as a result, their convictions were overturned.  You may or may not be able to get yours overturned, but it may be your last option to undo the effects of the charge.  To get a proper review, obtain copies of all of your original paperwork before you meet with the attorney.  Without the paperwork, their assessment of your case will be much more limited.