If you do not like your court appointed attorney you are free to hire your own attorney at any time. IT sounds like you are charged with criminal mischief. The punishment level is based on the amount of loss claimed by the owner of the damage property. If the value of the damage is more than $1500 then it can be a felony charge if less than $1500 then it can be charged as a misdemeanor. The court appointed attorney is correct the State can proceed to trial based only on the testimony of one person. At trial the State has to prove to a jury beyond a reasonable doubt that you committed the crime. Also if you are a permanent resident you should consult an immigration attorney because this criminal case can have consequences to your status and carry possible deportation.
Are you charged in a misdemeanor court or a felony court? (The felony courts in Harris County have 3 digits; misdemeanor courts have 1 or 12 digits.)
You can hire a lawyer if you are not pleased with the work your court appointed lawyer is doing. However, it sounds like the lawyer is doing what they are supposed to do. They are relaying the information that the DA's office is giving to them as well as obtaining the evidence in the case.
If you did not commit the crime, then you have a right to a jury trial. You can call witnesses to testify on your behalf as well as have your accusers cross examined by your lawyer.
Whether you are an American citizen matters to some judges; some will not put a person who is not a citizen on probation. And, you can be deported for some offenses, especially felony offenses.