You should immediately file a custody suit (SAPCR) and set a Temporary Orders hearing. At that hearing you can get a DNA test on the spot, and then if the results are positive you can come back to court to deal with custody. It will be difficult to get custody of a newborn, barring very strong facts against the Mom. I don't think the fact that she's living with another man or was considering an abortion are very relevant. The suicidal behavior and her own mother's concerns might be.
This is a complex matter, and you will need to hire the best custody lawyer that you can find. I have posted a link to some further information below.
This information is not legal advice and does not form an attorney-client relationship.
You live in Harris County. Depending on the court and the facts of your case, you could end up with custody. The burden will be on you to prove that the child is in danger.
Quite frankly, the fact that the mother considered having an abortion is irrelevant. I would not bother bringing that up in court. The baby is born now and that is all that matters.
Her mental frame of mind is what is important now.
Hire a tough, smart attorney now.
If you want a referral to an attorney, contact me.
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