How can I obtain guardianship of my grandson, whose parents have literally abandoned and left with me?

You need to file a custody suit, which in Texas is called a "Suit Affecting Parent Child Relationship." (SAPCR).  You can allege abandonment as a basis for obtaining custody.  You want custody, not just guardianship, because you need to have the right to consent to medical treatment, enroll the child in school, or add the child to an insurance policy.  You get much more flexibility with custody.  Guardianship is really applicable to the management of assets-- not necessarily the parenting of a child. 

A SAPCR starts with the filing of a petition.  If you know where the parents live, then they will need to be served with notice of the suit.  (If they've abandoned the child and don't want the child back, they may actually agree to the suit and waive service).  If you have no idea of where they are at, then you can have them served by a process called "publication."  After the filing, you need to request a final hearing with the judge to have you named primary managing conservator. 

If you can find an attorney to help you, the process will go a bit quicker and the will insure that the paperwork is in proper order. More and more attorneys are accepting payment plans and credit cards.... so shop around to find one that you are comfortable with, that knows family law, and offers a payment structure that works with your current financies.  If you have the time, talk to more than one so that you can get a better feel for what you want-- you really need a person who will be diligent and who will take the time to listen to your concerns for your grandson.