A more reliable source of info is this documentary first shown 5 years ago and repeated last night on TV in the USA. It is President Bush's own account of events.
https://www.youtube.com/watch?v=ke_OgE_V6tQ
Air Force One did not have to land for fuel. President Bush wanted to return to Washington DC ASAP but the Secret Service advised against it for obvious reasons until the situation was more clear regarding any further threats.
After getting airborne from Sarasota and circling for a while over the Gulf of Mexico, it then proceeded to Barksdale AFB, Louisiana. President Bush wanted to pre-record a video address to the Nation for later release. Coincidentally, Barksdale AFB was on exercise at the time.
It then proceeded to Offutt AFB, Nebraska for a variety of reasons - the availability of a secure bunker and secure video conferencing facilities at Strategic Command.
Finally, it got airborne from Offutt AFB for Andrews AFB from where the President was transported to the White House aboard Marine One.