Texas (and most states) will instantly throw your butt in jail until you post bail on a bench warrant. Immigration is not problem but if ANY Texas LEO runs you ID for ANY reason you will be instantly arrested, hauled down to the local jail and held until you post bail.
Funny story about my father. He once was stopped in his boat by lake patrol in Arizona for towing a water skier without an observer. The law says you must have a second adult in the boat when towing a water skier and he was towing a friend by himself. The lake patrol wrote him a ticket. It got wet later when he hauled out his boat and was illegible. The lake was far from his home and he never figured on going back to that lake again anyway. He just forgot about the whole thing.
Fully 5 years later he was pulled over for a brake light out in Phoenix. A bench warrant came up. He was arrested and hauled down town. The rural county judge that issued the bench warrant had failed to set a bail amount. Phoenix could not allow him to pay bail when it was the rural county judge issued the warrant, it was Friday, the judge would not be back until Monday to set bail......... so that is how dad spent the weekend in jail for towing a water skier without an observer. Monday morning it turned out bail was like $20-$30 and the fine was eventually waived entirely in lieu of time served.