I am a PT, but I do not diagnose via the internet

Having said that, everyone is built and moves a bit differently. "Tucking your tail" may be your PT's way of getting you to engage your transverse abdominus. There are many ways to skin that cat, and the Foundation program is very solid and gets you to engage very well. Essentially Foundation teaches you to stabilize your spine in a neutral (for you, which varies from person to person) position while you complete tasks. Very safe, and an excellent way to move.
Pain is by no means necessary to see improvement. Talk to your PT, let them know your concerns, and tell them about the benefit you feel with Foundation. It's an very well thought out and good program.
Good luck!