The machines are shielded so TSA people are protected.
X ray damage is cumulative, ie one scan will not produce fog, but additional scans will. It is like exposing a frame 5 stops under. it will be blank. Do it 5 times, you get fog. Also areas of low exposure are already past threshold, so one scan will do some damage in that area.
Overall fog is difficult to see unless you have a comparison. Generally it lowers contrast. Claims it can not be seen have to be discounted.
Modern x ray in developed countries do less damage than what may be found in third world countries.
Baggage x ray is much stronger WILL damage the film if that luggage is selected, but every piece is not checked.
Kodak used to explain all this on their website. No idea if still there.
Buy film here, ship UPS or Fed Ex to your hotel. Have it marked no ray and pray they can read. Develop before return. Or buy there, process there.
Digital is much less hassle. Rent one.