Can you get one? yes. Will you be arrested on the spot, most likely if the warrent is in the system. I would suggest that you clear up the warrent in Michigan. If you are stoppped or arrested you will sit in jail until they can transport back to Michigan. Then you will sit there until your Court date. Best to hire a lawyer in Michigan and clean it up then risk wasting day in jail waiting to clear it up.
If you go to get a Texas identification card, they will run your criminal record. If the misdemeanor warrant appears, which it should, and the police verify that it is an active warrant, you will be arrested. You will be brought to court in the county in which you are arrested and asked whether you want to waive extradition or fight it. If you waive it, notice will be given to Michigan and they will have 10 business days to come get you or you will be released. However, the warrant will still be pending & the process will occur over and over until you go to Michigan and settle the situation.
If you do not waive extradition, Texas will request confirmation from Michigan's governor, who will send it to Texas. The only question is whether there is a warrant and whether it is for you. A hearing will be held. Then, the same thing - they will have 10 days to come get you or you will be released & the process can begin again at any time.
For many reasons, including that they will look more favorably on you in Michigan if you turn yourself in, you should seek counsel in Michigan and take care of that business and not let a misdemeanor haunt you for the rest of your life. It is definitely NOT worth it!