You don't need a DNA test, they will know you are the mom. There is one DNA test, with a sample from the baby and from dad, and you don't have to have one if he acknowledges paternity. If you file a paternity action, then he will have an opportunity to establish custody and time sharing at your expense. If you wait til the child is born and file for child support only through the Department of Revenue, it does not establish any rights for him. As long as you are the unmarried mother with no court order regarding time sharing or custody, then you have all the rights. So you really have nothing to gain that I can see by filing now. He should be contributing to your expenses and care during the pregnancy if he want to be part of his child's life. I can't tell from your question if he wants to be and you are discouraging it, or if he is voluntarily choosing not to be involved, and if so, what exactly you want and/or fear from him.
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The paternity statute says that a paternity action may be filed by a woman who was pregnant, therefore you do not have to wait until the baby is born. You may have to wait if the father contest paternity for a paternity judgment, because the taking of the DNA test before birth is more difficult. also, and as difficult as it is to say, it is possible that you suffer a miscarriage, making the spending of the money for naught. But you may want to go ahead and get the ball rolling. I would probably recommend that you wait until you are little closer to term.
The statute has a fee shifting provision, so the court may award you your attorneys fees. A lot depends upon the father's financial condition. The Department of revenue will assist in the collection of child support and birth related expenses, but they will not handle issues of parental responsibility. You are better off hiring a lawyer, if you can afford and/or if you can find an attorney who will work with you on the fees, with the expectation that the father will pay all or some of the fees.
Don't forget to keep good records regarding any birth related expenses you incur.
The above is for informational and educational purposes only, does not establish an attorney client relationship, and cannot be relied upon as legal advice. The only legal advice offered is to have this matter reviewed and discussed by a competent attorney of your own choice.