So NASA probably didn’t look closely into the engineering, in particular when launch is tomorrow.
Yes, they did. NASA had been told by Thiokol the previous summer about the O-ring issue and that it could cause the loss of the Shuttle--and ignored the recommendation to ground the Shuttle until the issue was fixed. The night before the launch there was a conference call where the Thiokol engineers recommended not launching. Detailed engineering information was presented on that call--and it was information that had already been presented to NASA previously. NASA knew the engineering information and recommendation. They chose to ignore it.
The form he talked about was one that, if not signed, would mean that the launch would not happen. I can't remember if it was an internal form or not, but it doesn't really matter in that context.
Since NASA needed that form signed, he was under intense pressure to actually sign it both by NASA and his company. Someone else from the company not on site signed it.