how not to burn bridges between two academic jobs?
What stood out to me is:


  This same institution has already booked me for some partial hour lecturers in a couple of courses.


Have you, in any time, either written or orally, communicated with institute B indicating that you will take up the position (even just the part time one)? If so, the polite thing would be to honor this commitment with Institute B, even just for one term (if you really want to work at Institute A, you can try to talk to them to see if they can arrange for the offer to be deferred for a year or a term). 

If you have not indicated to institute B that you would take up the position, and they, without your knowledge, advertised you as a part-time lecturer for their courses, I would be very, very wary of joining their faculty. It is simply not the case that a department can strong-arm a potential hire into working there just by listing his or her name on a website. 



Aside from the above: the job market is competitive. It is expected that you will be looking out for your best interest. As long as you have not made any formal commitment to either of the institutes, you are free to choose which one to affiliate yourself with in the future. If you really prefer the offer given by Institute A, you should just be honest and reply to Institute B and say that Institute A has given you a better offer and you regret that you will not be able to work for Institute B. 

This really should not be a situation that counts as "burning bridges". From your description it sounds like just the normal competitive hiring process. If the administrators at Institute B are the type that will hold your declining of their offer against you in the future, I really doubt you will want to work for them anyway. (That just doesn't sound like a healthy work environment.)