If Iwas charged with felony assault which was later lowered to misdemeanor assault, can I get this removed from my criminal record?

You have several options depending on a couple of different factors and what exactly you want to get removed from your record.  If you pled guilty or no contest to the misd assault, expungement is going to be harder because it's already part of the court's records.  Pardons are extremely rare in Texas, but if you have unique circumstances, it may be a long shot option.  Your best option is a petition for non-disclosure if you were placed on deferred adjudication.  The result of this petition is that the courts and law enforcment will still know about your assault charge, but it won't be disclosed to employers... which is really what your goal is.  As far as your felony charge goes, you will actually have more options.  If the felony never went forward, then you will be eligible for an expunction.  Texas recently relaxed some of the rules regarding expunction.  There are certain timelines you need to wait until you are eligible, but if you can wait them out, this will be your best option.  For either the misdemeanor or the felony, when and how you can use these options will depend on more facts specific to your case.  Considering that these charges are hindering employment opportunities, investing in a consultation with an attorney that routinely does expunctions is a good investment.